// File: contracts/CurveUADIncentive.sol



// SPDX-License-Identifier: MIT
pragma solidity ^0.8.3;

import "./UbiquityAlgorithmicDollarManager.sol";
import "./interfaces/IUbiquityGovernance.sol";
import "./interfaces/IIncentive.sol";
import "./TWAPOracle.sol";
import "./UbiquityAlgorithmicDollar.sol";
import "./libs/ABDKMathQuad.sol";

/// @title Uniswap trading incentive contract
/// @author uAD Protocol
/// @dev incentives
contract CurveUADIncentive is IIncentive {
    using ABDKMathQuad for uint256;
    using ABDKMathQuad for bytes16;
    UbiquityAlgorithmicDollarManager public manager;
    bool public isSellPenaltyOn = true;
    bool public isBuyIncentiveOn = true;
    bytes16 private immutable _one = (uint256(1 ether)).fromUInt();
    mapping(address => bool) private _exempt;
    event ExemptAddressUpdate(address indexed _account, bool _isExempt);
    modifier onlyAdmin() {
        require(
            manager.hasRole(manager.INCENTIVE_MANAGER_ROLE(), msg.sender),
            "CurveIncentive: not admin"
        );
        _;
    }
    modifier onlyUAD() {
        require(
            msg.sender == manager.dollarTokenAddress(),
            "CurveIncentive: Caller is not uAD"
        );
        _;
    }

    /// @notice CurveIncentive constructor
    /// @param _manager uAD Manager
    constructor(address _manager) {
        manager = UbiquityAlgorithmicDollarManager(_manager);
    }

    function incentivize(
        address sender,
        address receiver,
        address,
        uint256 amountIn
    ) external override onlyUAD {
        require(sender != receiver, "CurveIncentive: cannot send self");

        if (sender == manager.stableSwapMetaPoolAddress()) {
            _incentivizeBuy(receiver, amountIn);
        }

        if (receiver == manager.stableSwapMetaPoolAddress()) {
            _incentivizeSell(sender, amountIn);
        }
    }

    /// @notice set an address to be exempted from Uniswap trading incentives
    /// @param account the address to update
    /// @param isExempt a flag for whether to exempt or unexempt
    function setExemptAddress(address account, bool isExempt)
        external
        onlyAdmin
    {
        _exempt[account] = isExempt;
        emit ExemptAddressUpdate(account, isExempt);
    }

    /// @notice switch the sell penalty
    function switchSellPenalty() external onlyAdmin {
        isSellPenaltyOn = !isSellPenaltyOn;
    }

    /// @notice switch the buy incentive
    function switchBuyIncentive() external onlyAdmin {
        isBuyIncentiveOn = !isBuyIncentiveOn;
    }

    /// @notice returns true if account is marked as exempt
    function isExemptAddress(address account) public view returns (bool) {
        return _exempt[account];
    }

    function _incentivizeBuy(address target, uint256 amountIn) internal {
        _updateOracle();

        if (isExemptAddress(target) || !isBuyIncentiveOn) {
            return;
        }

        uint256 incentive = _getPercentDeviationFromUnderPeg(amountIn);
        /* swapping 3CRV (or underlying) for uAD (aka buying uAD) will mint x% of uGOV.
             Where x = (1- TWAP_Price) * amountIn.
            E.g. uAD = 0.8, you buy 1000 uAD, you get (1-0.8)*1000 = 200 uGOV */

        if (incentive != 0) {
            // this means CurveIncentive should be a minter of UGOV
            IUbiquityGovernance(manager.governanceTokenAddress()).mint(
                target,
                incentive
            );
        }
    }

    /// @notice returns the percentage of deviation from the peg multiplied by amount
    //          when uAD is <1$
    function _getPercentDeviationFromUnderPeg(uint256 amount)
        internal
        returns (uint256)
    {
        _updateOracle();
        uint256 curPrice = _getTWAPPrice();
        if (curPrice >= 1 ether) {
            return 0;
        }

        uint256 res =
            _one
                .sub(curPrice.fromUInt())
                .mul((amount.fromUInt().div(_one)))
                .toUInt();
        // returns (1- TWAP_Price) * amount.
        return res;
    }

    function _incentivizeSell(address target, uint256 amount) internal {
        _updateOracle();
        if (isExemptAddress(target) || !isSellPenaltyOn) {
            return;
        }

        /*
        WARNING
        From curve doc :Tokens that take a fee upon a successful transfer may cause the curve pool
        to break or act in unexpected ways.
        fei does it differently because they can make sure only one contract has the ability to sell
        uAD and they control the whole liquidity pool on uniswap.
        here to avoid problem with the curve pool we execute the transfer as specified and then we
        take the penalty so if penalty + amount > balance then we revert
        swapping uAD for 3CRV (or underlying) (aka selling uAD) will burn x% of uAD
        Where x = (1- TWAP_Price) *100.
        */

        uint256 penalty = _getPercentDeviationFromUnderPeg(amount);
        if (penalty != 0) {
            require(penalty < amount, "Dollar: burn exceeds trade size");

            require(
                UbiquityAlgorithmicDollar(manager.dollarTokenAddress())
                    .balanceOf(target) >= penalty + amount,
                "Dollar: balance too low to get penalized"
            );
            UbiquityAlgorithmicDollar(manager.dollarTokenAddress()).burnFrom(
                target,
                penalty
            ); // burn from the recipient
        }
    }

    function _updateOracle() internal {
        TWAPOracle(manager.twapOracleAddress()).update();
    }

    function _getTWAPPrice() internal view returns (uint256) {
        return
            TWAPOracle(manager.twapOracleAddress()).consult(
                manager.dollarTokenAddress()
            );
    }
}

// File: contracts/UbiquityAlgorithmicDollarManager.sol



// SPDX-License-Identifier: MIT
pragma solidity ^0.8.3;

import "@openzeppelin/contracts/access/AccessControl.sol";
import "@openzeppelin/contracts/token/ERC20/ERC20.sol";
import "@openzeppelin/contracts/token/ERC20/IERC20.sol";
import "@openzeppelin/contracts/token/ERC20/utils/SafeERC20.sol";

import "./interfaces/IUbiquityAlgorithmicDollar.sol";
import "./interfaces/ICurveFactory.sol";
import "./interfaces/IMetaPool.sol";

import "./TWAPOracle.sol";

/// @title A central config for the uAD system. Also acts as a central
/// access control manager.
/// @notice For storing constants. For storing variables and allowing them to
/// be changed by the admin (governance)
/// @dev This should be used as a central access control manager which other
/// contracts use to check permissions
contract UbiquityAlgorithmicDollarManager is AccessControl {
    using SafeERC20 for IERC20;

    bytes32 public constant UBQ_MINTER_ROLE = keccak256("UBQ_MINTER_ROLE");
    bytes32 public constant UBQ_BURNER_ROLE = keccak256("UBQ_BURNER_ROLE");

    bytes32 public constant PAUSER_ROLE = keccak256("PAUSER_ROLE");
    bytes32 public constant COUPON_MANAGER_ROLE = keccak256("COUPON_MANAGER");
    bytes32 public constant BONDING_MANAGER_ROLE = keccak256("BONDING_MANAGER");
    bytes32 public constant INCENTIVE_MANAGER_ROLE =
        keccak256("INCENTIVE_MANAGER");
    bytes32 public constant UBQ_TOKEN_MANAGER_ROLE =
        keccak256("UBQ_TOKEN_MANAGER_ROLE");
    address public twapOracleAddress;
    address public debtCouponAddress;
    address public dollarTokenAddress; // uAD
    address public couponCalculatorAddress;
    address public dollarMintingCalculatorAddress;
    address public bondingShareAddress;
    address public bondingContractAddress;
    address public stableSwapMetaPoolAddress;
    address public curve3PoolTokenAddress; // 3CRV
    address public treasuryAddress;
    address public governanceTokenAddress; // uGOV
    address public sushiSwapPoolAddress; // sushi pool uAD-uGOV
    address public masterChefAddress;
    address public formulasAddress;
    address public autoRedeemTokenAddress; // uAR
    address public uarCalculatorAddress; // uAR calculator

    //key = address of couponmanager, value = excessdollardistributor
    mapping(address => address) private _excessDollarDistributors;

    modifier onlyAdmin() {
        require(
            hasRole(DEFAULT_ADMIN_ROLE, msg.sender),
            "uADMGR: Caller is not admin"
        );
        _;
    }

    constructor(address _admin) {
        _setupRole(DEFAULT_ADMIN_ROLE, _admin);
        _setupRole(UBQ_MINTER_ROLE, _admin);
        _setupRole(PAUSER_ROLE, _admin);
        _setupRole(COUPON_MANAGER_ROLE, _admin);
        _setupRole(BONDING_MANAGER_ROLE, _admin);
        _setupRole(INCENTIVE_MANAGER_ROLE, _admin);
        _setupRole(UBQ_TOKEN_MANAGER_ROLE, address(this));
    }

    // TODO Add a generic setter for extra addresses that needs to be linked
    function setTwapOracleAddress(address _twapOracleAddress)
        external
        onlyAdmin
    {
        twapOracleAddress = _twapOracleAddress;
        // to be removed

        TWAPOracle oracle = TWAPOracle(twapOracleAddress);
        oracle.update();
    }

    function setuARTokenAddress(address _uarTokenAddress) external onlyAdmin {
        autoRedeemTokenAddress = _uarTokenAddress;
    }

    function setDebtCouponAddress(address _debtCouponAddress)
        external
        onlyAdmin
    {
        debtCouponAddress = _debtCouponAddress;
    }

    function setIncentiveToUAD(address _account, address _incentiveAddress)
        external
        onlyAdmin
    {
        IUbiquityAlgorithmicDollar(dollarTokenAddress).setIncentiveContract(
            _account,
            _incentiveAddress
        );
    }

    function setDollarTokenAddress(address _dollarTokenAddress)
        external
        onlyAdmin
    {
        dollarTokenAddress = _dollarTokenAddress;
    }

    function setGovernanceTokenAddress(address _governanceTokenAddress)
        external
        onlyAdmin
    {
        governanceTokenAddress = _governanceTokenAddress;
    }

    function setSushiSwapPoolAddress(address _sushiSwapPoolAddress)
        external
        onlyAdmin
    {
        sushiSwapPoolAddress = _sushiSwapPoolAddress;
    }

    function setUARCalculatorAddress(address _uarCalculatorAddress)
        external
        onlyAdmin
    {
        uarCalculatorAddress = _uarCalculatorAddress;
    }

    function setCouponCalculatorAddress(address _couponCalculatorAddress)
        external
        onlyAdmin
    {
        couponCalculatorAddress = _couponCalculatorAddress;
    }

    function setDollarMintingCalculatorAddress(
        address _dollarMintingCalculatorAddress
    ) external onlyAdmin {
        dollarMintingCalculatorAddress = _dollarMintingCalculatorAddress;
    }

    function setExcessDollarsDistributor(
        address debtCouponManagerAddress,
        address excessCouponDistributor
    ) external onlyAdmin {
        _excessDollarDistributors[
            debtCouponManagerAddress
        ] = excessCouponDistributor;
    }

    function setMasterChefAddress(address _masterChefAddress)
        external
        onlyAdmin
    {
        masterChefAddress = _masterChefAddress;
    }

    function setFormulasAddress(address _formulasAddress) external onlyAdmin {
        formulasAddress = _formulasAddress;
    }

    function setBondingShareAddress(address _bondingShareAddress)
        external
        onlyAdmin
    {
        bondingShareAddress = _bondingShareAddress;
    }

    function setStableSwapMetaPoolAddress(address _stableSwapMetaPoolAddress)
        external
        onlyAdmin
    {
        stableSwapMetaPoolAddress = _stableSwapMetaPoolAddress;
    }

    /**
    @notice set the bonding bontract smart contract address
    @dev bonding contract participants deposit  curve LP token
         for a certain duration to earn uGOV and more curve LP token
    @param _bondingContractAddress bonding contract address
     */
    function setBondingContractAddress(address _bondingContractAddress)
        external
        onlyAdmin
    {
        bondingContractAddress = _bondingContractAddress;
    }

    /**
    @notice set the treasury address
    @dev the treasury fund is used to maintain the protocol
    @param _treasuryAddress treasury fund address
     */
    function setTreasuryAddress(address _treasuryAddress) external onlyAdmin {
        treasuryAddress = _treasuryAddress;
    }

    /**
    @notice deploy a new Curve metapools for uAD Token uAD/3Pool
    @dev  From the curve documentation for uncollateralized algorithmic
    stablecoins amplification should be 5-10
    @param _curveFactory MetaPool factory address
    @param _crvBasePool Address of the base pool to use within the new metapool.
    @param _crv3PoolTokenAddress curve 3Pool token Address
    @param _amplificationCoefficient amplification coefficient. The smaller
     it is the closer to a constant product we are.
    @param _fee Trade fee, given as an integer with 1e10 precision.
    */
    function deployStableSwapPool(
        address _curveFactory,
        address _crvBasePool,
        address _crv3PoolTokenAddress,
        uint256 _amplificationCoefficient,
        uint256 _fee
    ) external onlyAdmin {
        // Create new StableSwap meta pool (uAD <-> 3Crv)
        address metaPool =
            ICurveFactory(_curveFactory).deploy_metapool(
                _crvBasePool,
                ERC20(dollarTokenAddress).name(),
                ERC20(dollarTokenAddress).symbol(),
                dollarTokenAddress,
                _amplificationCoefficient,
                _fee
            );
        stableSwapMetaPoolAddress = metaPool;

        // Approve the newly-deployed meta pool to transfer this contract's funds
        uint256 crv3PoolTokenAmount =
            IERC20(_crv3PoolTokenAddress).balanceOf(address(this));
        uint256 uADTokenAmount =
            IERC20(dollarTokenAddress).balanceOf(address(this));

        // safe approve revert if approve from non-zero to non-zero allowance
        IERC20(_crv3PoolTokenAddress).safeApprove(metaPool, 0);
        IERC20(_crv3PoolTokenAddress).safeApprove(
            metaPool,
            crv3PoolTokenAmount
        );

        IERC20(dollarTokenAddress).safeApprove(metaPool, 0);
        IERC20(dollarTokenAddress).safeApprove(metaPool, uADTokenAmount);

        // coin at index 0 is uAD and index 1 is 3CRV
        require(
            IMetaPool(metaPool).coins(0) == dollarTokenAddress &&
                IMetaPool(metaPool).coins(1) == _crv3PoolTokenAddress,
            "uADMGR: COIN_ORDER_MISMATCH"
        );
        // Add the initial liquidity to the StableSwap meta pool
        uint256[2] memory amounts =
            [
                IERC20(dollarTokenAddress).balanceOf(address(this)),
                IERC20(_crv3PoolTokenAddress).balanceOf(address(this))
            ];

        // set curve 3Pool address
        curve3PoolTokenAddress = _crv3PoolTokenAddress;
        IMetaPool(metaPool).add_liquidity(amounts, 0, msg.sender);
    }

    function getExcessDollarsDistributor(address _debtCouponManagerAddress)
        external
        view
        returns (address)
    {
        return _excessDollarDistributors[_debtCouponManagerAddress];
    }
}

// File: contracts/interfaces/IUbiquityGovernance.sol



// SPDX-License-Identifier: Apache-2.0
pragma solidity ^0.8.3;

import "./IERC20Ubiquity.sol";

/// @title uGOV governance token interface
/// @author Ubiquity Algorithmic Dollar
// solhint-disable-next-line no-empty-blocks
interface IUbiquityGovernance is IERC20Ubiquity {

}

// File: contracts/interfaces/IIncentive.sol



// SPDX-License-Identifier: MIT
pragma solidity ^0.8.3;

/// @title incentive contract interface
/// @notice Called by uAD token contract when transferring with an incentivized address
/// @dev should be appointed as a Minter or Burner as needed
interface IIncentive {
    /// @notice apply incentives on transfer
    /// @param sender the sender address of uAD
    /// @param receiver the receiver address of uAD
    /// @param operator the operator (msg.sender) of the transfer
    /// @param amount the amount of uAD transferred
    function incentivize(
        address sender,
        address receiver,
        address operator,
        uint256 amount
    ) external;
}

// File: contracts/TWAPOracle.sol



// SPDX-License-Identifier: Apache-2.0
pragma solidity ^0.8.3;

import "./interfaces/IMetaPool.sol";

contract TWAPOracle {
    address public immutable pool;
    address public immutable token0;
    address public immutable token1;
    uint256 public price0Average;
    uint256 public price1Average;
    uint256 public pricesBlockTimestampLast;
    uint256[2] public priceCumulativeLast;

    constructor(
        address _pool,
        address _uADtoken0,
        address _curve3CRVtoken1
    ) {
        pool = _pool;
        // coin at index 0 is uAD and index 1 is 3CRV
        require(
            IMetaPool(_pool).coins(0) == _uADtoken0 &&
                IMetaPool(_pool).coins(1) == _curve3CRVtoken1,
            "TWAPOracle: COIN_ORDER_MISMATCH"
        );

        token0 = _uADtoken0;
        token1 = _curve3CRVtoken1;

        uint256 _reserve0 = uint112(IMetaPool(_pool).balances(0));
        uint256 _reserve1 = uint112(IMetaPool(_pool).balances(1));

        // ensure that there's liquidity in the pair
        require(_reserve0 != 0 && _reserve1 != 0, "TWAPOracle: NO_RESERVES");
        // ensure that pair balance is perfect
        require(_reserve0 == _reserve1, "TWAPOracle: PAIR_UNBALANCED");
        priceCumulativeLast = IMetaPool(_pool).get_price_cumulative_last();
        pricesBlockTimestampLast = IMetaPool(_pool).block_timestamp_last();

        price0Average = 1 ether;
        price1Average = 1 ether;
    }

    // calculate average price
    function update() external {
        (uint256[2] memory priceCumulative, uint256 blockTimestamp) =
            _currentCumulativePrices();

        if (blockTimestamp - pricesBlockTimestampLast > 0) {
            // get the balances between now and the last price cumulative snapshot
            uint256[2] memory twapBalances =
                IMetaPool(pool).get_twap_balances(
                    priceCumulativeLast,
                    priceCumulative,
                    blockTimestamp - pricesBlockTimestampLast
                );

            // price to exchange amounIn uAD to 3CRV based on TWAP
            price0Average = IMetaPool(pool).get_dy(0, 1, 1 ether, twapBalances);
            // price to exchange amounIn 3CRV to uAD  based on TWAP
            price1Average = IMetaPool(pool).get_dy(1, 0, 1 ether, twapBalances);
            // we update the priceCumulative
            priceCumulativeLast = priceCumulative;
            pricesBlockTimestampLast = blockTimestamp;
        }
    }

    // note this will always return 0 before update has been called successfully
    // for the first time.
    function consult(address token) external view returns (uint256 amountOut) {
        if (token == token0) {
            // price to exchange 1 uAD to 3CRV based on TWAP
            amountOut = price0Average;
        } else {
            require(token == token1, "TWAPOracle: INVALID_TOKEN");
            // price to exchange 1 3CRV to uAD  based on TWAP
            amountOut = price1Average;
        }
    }

    function _currentCumulativePrices()
        internal
        view
        returns (uint256[2] memory priceCumulative, uint256 blockTimestamp)
    {
        priceCumulative = IMetaPool(pool).get_price_cumulative_last();
        blockTimestamp = IMetaPool(pool).block_timestamp_last();
    }
}

// File: contracts/UbiquityAlgorithmicDollar.sol



// SPDX-License-Identifier: MIT
pragma solidity ^0.8.3;

import "./interfaces/IIncentive.sol";
import "./ERC20Ubiquity.sol";

contract UbiquityAlgorithmicDollar is ERC20Ubiquity {
    /// @notice get associated incentive contract, 0 address if N/A
    mapping(address => address) public incentiveContract;

    event IncentiveContractUpdate(
        address indexed _incentivized,
        address indexed _incentiveContract
    );

    constructor(address _manager)
        ERC20Ubiquity(_manager, "Ubiquity Algorithmic Dollar", "uAD")
    {} // solhint-disable-line no-empty-blocks

    /// @param account the account to incentivize
    /// @param incentive the associated incentive contract
    /// @notice only UAD manager can set Incentive contract
    function setIncentiveContract(address account, address incentive) external {
        require(
            ERC20Ubiquity.manager.hasRole(
                ERC20Ubiquity.manager.UBQ_TOKEN_MANAGER_ROLE(),
                msg.sender
            ),
            "Dollar: must have admin role"
        );

        incentiveContract[account] = incentive;
        emit IncentiveContractUpdate(account, incentive);
    }

    function _checkAndApplyIncentives(
        address sender,
        address recipient,
        uint256 amount
    ) internal {
        // incentive on sender
        address senderIncentive = incentiveContract[sender];
        if (senderIncentive != address(0)) {
            IIncentive(senderIncentive).incentivize(
                sender,
                recipient,
                msg.sender,
                amount
            );
        }

        // incentive on recipient
        address recipientIncentive = incentiveContract[recipient];
        if (recipientIncentive != address(0)) {
            IIncentive(recipientIncentive).incentivize(
                sender,
                recipient,
                msg.sender,
                amount
            );
        }

        // incentive on operator
        address operatorIncentive = incentiveContract[msg.sender];
        if (
            msg.sender != sender &&
            msg.sender != recipient &&
            operatorIncentive != address(0)
        ) {
            IIncentive(operatorIncentive).incentivize(
                sender,
                recipient,
                msg.sender,
                amount
            );
        }

        // all incentive, if active applies to every transfer
        address allIncentive = incentiveContract[address(0)];
        if (allIncentive != address(0)) {
            IIncentive(allIncentive).incentivize(
                sender,
                recipient,
                msg.sender,
                amount
            );
        }
    }

    function _transfer(
        address sender,
        address recipient,
        uint256 amount
    ) internal override {
        super._transfer(sender, recipient, amount);
        _checkAndApplyIncentives(sender, recipient, amount);
    }
}

// File: contracts/libs/ABDKMathQuad.sol



// SPDX-License-Identifier: BSD-4-Clause
/*
 * ABDK Math Quad Smart Contract Library.  Copyright © 2019 by ABDK Consulting.
 * Author: Mikhail Vladimirov <mikhail.vladimirov@gmail.com>
 */
pragma solidity ^0.8.0;

/**
 * Smart contract library of mathematical functions operating with IEEE 754
 * quadruple-precision binary floating-point numbers (quadruple precision
 * numbers).  As long as quadruple precision numbers are 16-bytes long, they are
 * represented by bytes16 type.
 */
library ABDKMathQuad {
    /*
     * 0.
     */
    bytes16 private constant _POSITIVE_ZERO =
        0x00000000000000000000000000000000;

    /*
     * -0.
     */
    bytes16 private constant _NEGATIVE_ZERO =
        0x80000000000000000000000000000000;

    /*
     * +Infinity.
     */
    bytes16 private constant _POSITIVE_INFINITY =
        0x7FFF0000000000000000000000000000;

    /*
     * -Infinity.
     */
    bytes16 private constant _NEGATIVE_INFINITY =
        0xFFFF0000000000000000000000000000;

    /*
     * Canonical NaN value.
     */
    bytes16 private constant NaN = 0x7FFF8000000000000000000000000000;

    /**
     * Convert signed 256-bit integer number into quadruple precision number.
     *
     * @param x signed 256-bit integer number
     * @return quadruple precision number
     */
    function fromInt(int256 x) internal pure returns (bytes16) {
        unchecked {
            if (x == 0) return bytes16(0);
            else {
                // We rely on overflow behavior here
                uint256 result = uint256(x > 0 ? x : -x);

                uint256 msb = mostSignificantBit(result);
                if (msb < 112) result <<= 112 - msb;
                else if (msb > 112) result >>= msb - 112;

                result =
                    (result & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF) |
                    ((16383 + msb) << 112);
                if (x < 0) result |= 0x80000000000000000000000000000000;

                return bytes16(uint128(result));
            }
        }
    }

    /**
     * Convert quadruple precision number into signed 256-bit integer number
     * rounding towards zero.  Revert on overflow.
     *
     * @param x quadruple precision number
     * @return signed 256-bit integer number
     */
    function toInt(bytes16 x) internal pure returns (int256) {
        unchecked {
            uint256 exponent = (uint128(x) >> 112) & 0x7FFF;

            require(exponent <= 16638); // Overflow
            if (exponent < 16383) return 0; // Underflow

            uint256 result =
                (uint256(uint128(x)) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF) |
                    0x10000000000000000000000000000;

            if (exponent < 16495) result >>= 16495 - exponent;
            else if (exponent > 16495) result <<= exponent - 16495;

            if (uint128(x) >= 0x80000000000000000000000000000000) {
                // Negative
                require(
                    result <=
                        0x8000000000000000000000000000000000000000000000000000000000000000
                );
                return -int256(result); // We rely on overflow behavior here
            } else {
                require(
                    result <=
                        0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
                );
                return int256(result);
            }
        }
    }

    /**
     * Convert unsigned 256-bit integer number into quadruple precision number.
     *
     * @param x unsigned 256-bit integer number
     * @return quadruple precision number
     */
    function fromUInt(uint256 x) internal pure returns (bytes16) {
        unchecked {
            if (x == 0) return bytes16(0);
            else {
                uint256 result = x;

                uint256 msb = mostSignificantBit(result);
                if (msb < 112) result <<= 112 - msb;
                else if (msb > 112) result >>= msb - 112;

                result =
                    (result & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF) |
                    ((16383 + msb) << 112);

                return bytes16(uint128(result));
            }
        }
    }

    /**
     * Convert quadruple precision number into unsigned 256-bit integer number
     * rounding towards zero.  Revert on underflow.  Note, that negative floating
     * point numbers in range (-1.0 .. 0.0) may be converted to unsigned integer
     * without error, because they are rounded to zero.
     *
     * @param x quadruple precision number
     * @return unsigned 256-bit integer number
     */
    function toUInt(bytes16 x) internal pure returns (uint256) {
        unchecked {
            uint256 exponent = (uint128(x) >> 112) & 0x7FFF;

            if (exponent < 16383) return 0; // Underflow

            require(uint128(x) < 0x80000000000000000000000000000000); // Negative

            require(exponent <= 16638); // Overflow
            uint256 result =
                (uint256(uint128(x)) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF) |
                    0x10000000000000000000000000000;

            if (exponent < 16495) result >>= 16495 - exponent;
            else if (exponent > 16495) result <<= exponent - 16495;

            return result;
        }
    }

    /**
     * Convert signed 128.128 bit fixed point number into quadruple precision
     * number.
     *
     * @param x signed 128.128 bit fixed point number
     * @return quadruple precision number
     */
    function from128x128(int256 x) internal pure returns (bytes16) {
        unchecked {
            if (x == 0) return bytes16(0);
            else {
                // We rely on overflow behavior here
                uint256 result = uint256(x > 0 ? x : -x);

                uint256 msb = mostSignificantBit(result);
                if (msb < 112) result <<= 112 - msb;
                else if (msb > 112) result >>= msb - 112;

                result =
                    (result & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF) |
                    ((16255 + msb) << 112);
                if (x < 0) result |= 0x80000000000000000000000000000000;

                return bytes16(uint128(result));
            }
        }
    }

    /**
     * Convert quadruple precision number into signed 128.128 bit fixed point
     * number.  Revert on overflow.
     *
     * @param x quadruple precision number
     * @return signed 128.128 bit fixed point number
     */
    function to128x128(bytes16 x) internal pure returns (int256) {
        unchecked {
            uint256 exponent = (uint128(x) >> 112) & 0x7FFF;

            require(exponent <= 16510); // Overflow
            if (exponent < 16255) return 0; // Underflow

            uint256 result =
                (uint256(uint128(x)) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF) |
                    0x10000000000000000000000000000;

            if (exponent < 16367) result >>= 16367 - exponent;
            else if (exponent > 16367) result <<= exponent - 16367;

            if (uint128(x) >= 0x80000000000000000000000000000000) {
                // Negative
                require(
                    result <=
                        0x8000000000000000000000000000000000000000000000000000000000000000
                );
                return -int256(result); // We rely on overflow behavior here
            } else {
                require(
                    result <=
                        0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
                );
                return int256(result);
            }
        }
    }

    /**
     * Convert signed 64.64 bit fixed point number into quadruple precision
     * number.
     *
     * @param x signed 64.64 bit fixed point number
     * @return quadruple precision number
     */
    function from64x64(int128 x) internal pure returns (bytes16) {
        unchecked {
            if (x == 0) return bytes16(0);
            else {
                // We rely on overflow behavior here
                uint256 result = uint128(x > 0 ? x : -x);

                uint256 msb = mostSignificantBit(result);
                if (msb < 112) result <<= 112 - msb;
                else if (msb > 112) result >>= msb - 112;

                result =
                    (result & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF) |
                    ((16319 + msb) << 112);
                if (x < 0) result |= 0x80000000000000000000000000000000;

                return bytes16(uint128(result));
            }
        }
    }

    /**
     * Convert quadruple precision number into signed 64.64 bit fixed point
     * number.  Revert on overflow.
     *
     * @param x quadruple precision number
     * @return signed 64.64 bit fixed point number
     */
    function to64x64(bytes16 x) internal pure returns (int128) {
        unchecked {
            uint256 exponent = (uint128(x) >> 112) & 0x7FFF;

            require(exponent <= 16446); // Overflow
            if (exponent < 16319) return 0; // Underflow

            uint256 result =
                (uint256(uint128(x)) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF) |
                    0x10000000000000000000000000000;

            if (exponent < 16431) result >>= 16431 - exponent;
            else if (exponent > 16431) result <<= exponent - 16431;

            if (uint128(x) >= 0x80000000000000000000000000000000) {
                // Negative
                require(result <= 0x80000000000000000000000000000000);
                return -int128(int256(result)); // We rely on overflow behavior here
            } else {
                require(result <= 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
                return int128(int256(result));
            }
        }
    }

    /**
     * Convert octuple precision number into quadruple precision number.
     *
     * @param x octuple precision number
     * @return quadruple precision number
     */
    function fromOctuple(bytes32 x) internal pure returns (bytes16) {
        unchecked {
            bool negative =
                x &
                    0x8000000000000000000000000000000000000000000000000000000000000000 >
                    0;

            uint256 exponent = (uint256(x) >> 236) & 0x7FFFF;
            uint256 significand =
                uint256(x) &
                    0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

            if (exponent == 0x7FFFF) {
                if (significand > 0) return NaN;
                else return negative ? _NEGATIVE_INFINITY : _POSITIVE_INFINITY;
            }

            if (exponent > 278526)
                return negative ? _NEGATIVE_INFINITY : _POSITIVE_INFINITY;
            else if (exponent < 245649)
                return negative ? _NEGATIVE_ZERO : _POSITIVE_ZERO;
            else if (exponent < 245761) {
                significand =
                    (significand |
                        0x100000000000000000000000000000000000000000000000000000000000) >>
                    (245885 - exponent);
                exponent = 0;
            } else {
                significand >>= 124;
                exponent -= 245760;
            }

            uint128 result = uint128(significand | (exponent << 112));
            if (negative) result |= 0x80000000000000000000000000000000;

            return bytes16(result);
        }
    }

    /**
     * Convert quadruple precision number into octuple precision number.
     *
     * @param x quadruple precision number
     * @return octuple precision number
     */
    function toOctuple(bytes16 x) internal pure returns (bytes32) {
        unchecked {
            uint256 exponent = (uint128(x) >> 112) & 0x7FFF;

            uint256 result = uint128(x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

            if (exponent == 0x7FFF)
                exponent = 0x7FFFF; // Infinity or NaN
            else if (exponent == 0) {
                if (result > 0) {
                    uint256 msb = mostSignificantBit(result);
                    result =
                        (result << (236 - msb)) &
                        0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                    exponent = 245649 + msb;
                }
            } else {
                result <<= 124;
                exponent += 245760;
            }

            result |= exponent << 236;
            if (uint128(x) >= 0x80000000000000000000000000000000)
                result |= 0x8000000000000000000000000000000000000000000000000000000000000000;

            return bytes32(result);
        }
    }

    /**
     * Convert double precision number into quadruple precision number.
     *
     * @param x double precision number
     * @return quadruple precision number
     */
    function fromDouble(bytes8 x) internal pure returns (bytes16) {
        unchecked {
            uint256 exponent = (uint64(x) >> 52) & 0x7FF;

            uint256 result = uint64(x) & 0xFFFFFFFFFFFFF;

            if (exponent == 0x7FF)
                exponent = 0x7FFF; // Infinity or NaN
            else if (exponent == 0) {
                if (result > 0) {
                    uint256 msb = mostSignificantBit(result);
                    result =
                        (result << (112 - msb)) &
                        0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                    exponent = 15309 + msb;
                }
            } else {
                result <<= 60;
                exponent += 15360;
            }

            result |= exponent << 112;
            if (x & 0x8000000000000000 > 0)
                result |= 0x80000000000000000000000000000000;

            return bytes16(uint128(result));
        }
    }

    /**
     * Convert quadruple precision number into double precision number.
     *
     * @param x quadruple precision number
     * @return double precision number
     */
    function toDouble(bytes16 x) internal pure returns (bytes8) {
        unchecked {
            bool negative = uint128(x) >= 0x80000000000000000000000000000000;

            uint256 exponent = (uint128(x) >> 112) & 0x7FFF;
            uint256 significand = uint128(x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

            if (exponent == 0x7FFF) {
                if (significand > 0) return 0x7FF8000000000000;
                // NaN
                else
                    return
                        negative
                            ? bytes8(0xFFF0000000000000) // -Infinity
                            : bytes8(0x7FF0000000000000); // Infinity
            }

            if (exponent > 17406)
                return
                    negative
                        ? bytes8(0xFFF0000000000000) // -Infinity
                        : bytes8(0x7FF0000000000000);
            // Infinity
            else if (exponent < 15309)
                return
                    negative
                        ? bytes8(0x8000000000000000) // -0
                        : bytes8(0x0000000000000000);
            // 0
            else if (exponent < 15361) {
                significand =
                    (significand | 0x10000000000000000000000000000) >>
                    (15421 - exponent);
                exponent = 0;
            } else {
                significand >>= 60;
                exponent -= 15360;
            }

            uint64 result = uint64(significand | (exponent << 52));
            if (negative) result |= 0x8000000000000000;

            return bytes8(result);
        }
    }

    /**
     * Test whether given quadruple precision number is NaN.
     *
     * @param x quadruple precision number
     * @return true if x is NaN, false otherwise
     */
    function isNaN(bytes16 x) internal pure returns (bool) {
        unchecked {
            return
                uint128(x) & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF >
                0x7FFF0000000000000000000000000000;
        }
    }

    /**
     * Test whether given quadruple precision number is positive or negative
     * infinity.
     *
     * @param x quadruple precision number
     * @return true if x is positive or negative infinity, false otherwise
     */
    function isInfinity(bytes16 x) internal pure returns (bool) {
        unchecked {
            return
                uint128(x) & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF ==
                0x7FFF0000000000000000000000000000;
        }
    }

    /**
     * Calculate sign of x, i.e. -1 if x is negative, 0 if x if zero, and 1 if x
     * is positive.  Note that sign (-0) is zero.  Revert if x is NaN.
     *
     * @param x quadruple precision number
     * @return sign of x
     */
    function sign(bytes16 x) internal pure returns (int8) {
        unchecked {
            uint128 absoluteX = uint128(x) & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

            require(absoluteX <= 0x7FFF0000000000000000000000000000); // Not NaN

            if (absoluteX == 0) return 0;
            else if (uint128(x) >= 0x80000000000000000000000000000000)
                return -1;
            else return 1;
        }
    }

    /**
     * Calculate sign (x - y).  Revert if either argument is NaN, or both
     * arguments are infinities of the same sign.
     *
     * @param x quadruple precision number
     * @param y quadruple precision number
     * @return sign (x - y)
     */
    function cmp(bytes16 x, bytes16 y) internal pure returns (int8) {
        unchecked {
            uint128 absoluteX = uint128(x) & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

            require(absoluteX <= 0x7FFF0000000000000000000000000000); // Not NaN

            uint128 absoluteY = uint128(y) & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

            require(absoluteY <= 0x7FFF0000000000000000000000000000); // Not NaN

            // Not infinities of the same sign
            require(x != y || absoluteX < 0x7FFF0000000000000000000000000000);

            if (x == y) return 0;
            else {
                bool negativeX =
                    uint128(x) >= 0x80000000000000000000000000000000;
                bool negativeY =
                    uint128(y) >= 0x80000000000000000000000000000000;

                if (negativeX) {
                    if (negativeY) return absoluteX > absoluteY ? -1 : int8(1);
                    else return -1;
                } else {
                    if (negativeY) return 1;
                    else return absoluteX > absoluteY ? int8(1) : -1;
                }
            }
        }
    }

    /**
     * Test whether x equals y.  NaN, infinity, and -infinity are not equal to
     * anything.
     *
     * @param x quadruple precision number
     * @param y quadruple precision number
     * @return true if x equals to y, false otherwise
     */
    function eq(bytes16 x, bytes16 y) internal pure returns (bool) {
        unchecked {
            if (x == y) {
                return
                    uint128(x) & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF <
                    0x7FFF0000000000000000000000000000;
            } else return false;
        }
    }

    /**
     * Calculate x + y.  Special values behave in the following way:
     *
     * NaN + x = NaN for any x.
     * Infinity + x = Infinity for any finite x.
     * -Infinity + x = -Infinity for any finite x.
     * Infinity + Infinity = Infinity.
     * -Infinity + -Infinity = -Infinity.
     * Infinity + -Infinity = -Infinity + Infinity = NaN.
     *
     * @param x quadruple precision number
     * @param y quadruple precision number
     * @return quadruple precision number
     */
    function add(bytes16 x, bytes16 y) internal pure returns (bytes16) {
        unchecked {
            uint256 xExponent = (uint128(x) >> 112) & 0x7FFF;
            uint256 yExponent = (uint128(y) >> 112) & 0x7FFF;

            if (xExponent == 0x7FFF) {
                if (yExponent == 0x7FFF) {
                    if (x == y) return x;
                    else return NaN;
                } else return x;
            } else if (yExponent == 0x7FFF) return y;
            else {
                bool xSign = uint128(x) >= 0x80000000000000000000000000000000;
                uint256 xSignifier =
                    uint128(x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                if (xExponent == 0) xExponent = 1;
                else xSignifier |= 0x10000000000000000000000000000;

                bool ySign = uint128(y) >= 0x80000000000000000000000000000000;
                uint256 ySignifier =
                    uint128(y) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                if (yExponent == 0) yExponent = 1;
                else ySignifier |= 0x10000000000000000000000000000;

                if (xSignifier == 0)
                    return y == _NEGATIVE_ZERO ? _POSITIVE_ZERO : y;
                else if (ySignifier == 0)
                    return x == _NEGATIVE_ZERO ? _POSITIVE_ZERO : x;
                else {
                    int256 delta = int256(xExponent) - int256(yExponent);

                    if (xSign == ySign) {
                        if (delta > 112) return x;
                        else if (delta > 0) ySignifier >>= uint256(delta);
                        else if (delta < -112) return y;
                        else if (delta < 0) {
                            xSignifier >>= uint256(-delta);
                            xExponent = yExponent;
                        }

                        xSignifier += ySignifier;

                        if (xSignifier >= 0x20000000000000000000000000000) {
                            xSignifier >>= 1;
                            xExponent += 1;
                        }

                        if (xExponent == 0x7FFF)
                            return
                                xSign ? _NEGATIVE_INFINITY : _POSITIVE_INFINITY;
                        else {
                            if (xSignifier < 0x10000000000000000000000000000)
                                xExponent = 0;
                            else xSignifier &= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

                            return
                                bytes16(
                                    uint128(
                                        (
                                            xSign
                                                ? 0x80000000000000000000000000000000
                                                : 0
                                        ) |
                                            (xExponent << 112) |
                                            xSignifier
                                    )
                                );
                        }
                    } else {
                        if (delta > 0) {
                            xSignifier <<= 1;
                            xExponent -= 1;
                        } else if (delta < 0) {
                            ySignifier <<= 1;
                            xExponent = yExponent - 1;
                        }

                        if (delta > 112) ySignifier = 1;
                        else if (delta > 1)
                            ySignifier =
                                ((ySignifier - 1) >> uint256(delta - 1)) +
                                1;
                        else if (delta < -112) xSignifier = 1;
                        else if (delta < -1)
                            xSignifier =
                                ((xSignifier - 1) >> uint256(-delta - 1)) +
                                1;

                        if (xSignifier >= ySignifier) xSignifier -= ySignifier;
                        else {
                            xSignifier = ySignifier - xSignifier;
                            xSign = ySign;
                        }

                        if (xSignifier == 0) return _POSITIVE_ZERO;

                        uint256 msb = mostSignificantBit(xSignifier);

                        if (msb == 113) {
                            xSignifier =
                                (xSignifier >> 1) &
                                0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                            xExponent += 1;
                        } else if (msb < 112) {
                            uint256 shift = 112 - msb;
                            if (xExponent > shift) {
                                xSignifier =
                                    (xSignifier << shift) &
                                    0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                                xExponent -= shift;
                            } else {
                                xSignifier <<= xExponent - 1;
                                xExponent = 0;
                            }
                        } else xSignifier &= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

                        if (xExponent == 0x7FFF)
                            return
                                xSign ? _NEGATIVE_INFINITY : _POSITIVE_INFINITY;
                        else
                            return
                                bytes16(
                                    uint128(
                                        (
                                            xSign
                                                ? 0x80000000000000000000000000000000
                                                : 0
                                        ) |
                                            (xExponent << 112) |
                                            xSignifier
                                    )
                                );
                    }
                }
            }
        }
    }

    /**
     * Calculate x - y.  Special values behave in the following way:
     *
     * NaN - x = NaN for any x.
     * Infinity - x = Infinity for any finite x.
     * -Infinity - x = -Infinity for any finite x.
     * Infinity - -Infinity = Infinity.
     * -Infinity - Infinity = -Infinity.
     * Infinity - Infinity = -Infinity - -Infinity = NaN.
     *
     * @param x quadruple precision number
     * @param y quadruple precision number
     * @return quadruple precision number
     */
    function sub(bytes16 x, bytes16 y) internal pure returns (bytes16) {
        unchecked {return add(x, y ^ 0x80000000000000000000000000000000);}
    }

    /**
     * Calculate x * y.  Special values behave in the following way:
     *
     * NaN * x = NaN for any x.
     * Infinity * x = Infinity for any finite positive x.
     * Infinity * x = -Infinity for any finite negative x.
     * -Infinity * x = -Infinity for any finite positive x.
     * -Infinity * x = Infinity for any finite negative x.
     * Infinity * 0 = NaN.
     * -Infinity * 0 = NaN.
     * Infinity * Infinity = Infinity.
     * Infinity * -Infinity = -Infinity.
     * -Infinity * Infinity = -Infinity.
     * -Infinity * -Infinity = Infinity.
     *
     * @param x quadruple precision number
     * @param y quadruple precision number
     * @return quadruple precision number
     */
    function mul(bytes16 x, bytes16 y) internal pure returns (bytes16) {
        unchecked {
            uint256 xExponent = (uint128(x) >> 112) & 0x7FFF;
            uint256 yExponent = (uint128(y) >> 112) & 0x7FFF;

            if (xExponent == 0x7FFF) {
                if (yExponent == 0x7FFF) {
                    if (x == y)
                        return x ^ (y & 0x80000000000000000000000000000000);
                    else if (x ^ y == 0x80000000000000000000000000000000)
                        return x | y;
                    else return NaN;
                } else {
                    if (y & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF == 0) return NaN;
                    else return x ^ (y & 0x80000000000000000000000000000000);
                }
            } else if (yExponent == 0x7FFF) {
                if (x & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF == 0) return NaN;
                else return y ^ (x & 0x80000000000000000000000000000000);
            } else {
                uint256 xSignifier =
                    uint128(x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                if (xExponent == 0) xExponent = 1;
                else xSignifier |= 0x10000000000000000000000000000;

                uint256 ySignifier =
                    uint128(y) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                if (yExponent == 0) yExponent = 1;
                else ySignifier |= 0x10000000000000000000000000000;

                xSignifier *= ySignifier;
                if (xSignifier == 0)
                    return
                        (x ^ y) & 0x80000000000000000000000000000000 > 0
                            ? _NEGATIVE_ZERO
                            : _POSITIVE_ZERO;

                xExponent += yExponent;

                uint256 msb =
                    xSignifier >=
                        0x200000000000000000000000000000000000000000000000000000000
                        ? 225
                        : xSignifier >=
                            0x100000000000000000000000000000000000000000000000000000000
                        ? 224
                        : mostSignificantBit(xSignifier);

                if (xExponent + msb < 16496) {
                    // Underflow
                    xExponent = 0;
                    xSignifier = 0;
                } else if (xExponent + msb < 16608) {
                    // Subnormal
                    if (xExponent < 16496) xSignifier >>= 16496 - xExponent;
                    else if (xExponent > 16496)
                        xSignifier <<= xExponent - 16496;
                    xExponent = 0;
                } else if (xExponent + msb > 49373) {
                    xExponent = 0x7FFF;
                    xSignifier = 0;
                } else {
                    if (msb > 112) xSignifier >>= msb - 112;
                    else if (msb < 112) xSignifier <<= 112 - msb;

                    xSignifier &= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

                    xExponent = xExponent + msb - 16607;
                }

                return
                    bytes16(
                        uint128(
                            uint128(
                                (x ^ y) & 0x80000000000000000000000000000000
                            ) |
                                (xExponent << 112) |
                                xSignifier
                        )
                    );
            }
        }
    }

    /**
     * Calculate x / y.  Special values behave in the following way:
     *
     * NaN / x = NaN for any x.
     * x / NaN = NaN for any x.
     * Infinity / x = Infinity for any finite non-negative x.
     * Infinity / x = -Infinity for any finite negative x including -0.
     * -Infinity / x = -Infinity for any finite non-negative x.
     * -Infinity / x = Infinity for any finite negative x including -0.
     * x / Infinity = 0 for any finite non-negative x.
     * x / -Infinity = -0 for any finite non-negative x.
     * x / Infinity = -0 for any finite non-negative x including -0.
     * x / -Infinity = 0 for any finite non-negative x including -0.
     *
     * Infinity / Infinity = NaN.
     * Infinity / -Infinity = -NaN.
     * -Infinity / Infinity = -NaN.
     * -Infinity / -Infinity = NaN.
     *
     * Division by zero behaves in the following way:
     *
     * x / 0 = Infinity for any finite positive x.
     * x / -0 = -Infinity for any finite positive x.
     * x / 0 = -Infinity for any finite negative x.
     * x / -0 = Infinity for any finite negative x.
     * 0 / 0 = NaN.
     * 0 / -0 = NaN.
     * -0 / 0 = NaN.
     * -0 / -0 = NaN.
     *
     * @param x quadruple precision number
     * @param y quadruple precision number
     * @return quadruple precision number
     */
    function div(bytes16 x, bytes16 y) internal pure returns (bytes16) {
        unchecked {
            uint256 xExponent = (uint128(x) >> 112) & 0x7FFF;
            uint256 yExponent = (uint128(y) >> 112) & 0x7FFF;

            if (xExponent == 0x7FFF) {
                if (yExponent == 0x7FFF) return NaN;
                else return x ^ (y & 0x80000000000000000000000000000000);
            } else if (yExponent == 0x7FFF) {
                if (y & 0x0000FFFFFFFFFFFFFFFFFFFFFFFFFFFF != 0) return NaN;
                else
                    return
                        _POSITIVE_ZERO |
                        ((x ^ y) & 0x80000000000000000000000000000000);
            } else if (y & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF == 0) {
                if (x & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF == 0) return NaN;
                else
                    return
                        _POSITIVE_INFINITY |
                        ((x ^ y) & 0x80000000000000000000000000000000);
            } else {
                uint256 ySignifier =
                    uint128(y) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                if (yExponent == 0) yExponent = 1;
                else ySignifier |= 0x10000000000000000000000000000;

                uint256 xSignifier =
                    uint128(x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                if (xExponent == 0) {
                    if (xSignifier != 0) {
                        uint256 shift = 226 - mostSignificantBit(xSignifier);

                        xSignifier <<= shift;

                        xExponent = 1;
                        yExponent += shift - 114;
                    }
                } else {
                    xSignifier =
                        (xSignifier | 0x10000000000000000000000000000) <<
                        114;
                }

                xSignifier = xSignifier / ySignifier;
                if (xSignifier == 0)
                    return
                        (x ^ y) & 0x80000000000000000000000000000000 > 0
                            ? _NEGATIVE_ZERO
                            : _POSITIVE_ZERO;

                assert(xSignifier >= 0x1000000000000000000000000000);

                uint256 msb =
                    xSignifier >= 0x80000000000000000000000000000
                        ? mostSignificantBit(xSignifier)
                        : xSignifier >= 0x40000000000000000000000000000
                        ? 114
                        : xSignifier >= 0x20000000000000000000000000000
                        ? 113
                        : 112;

                if (xExponent + msb > yExponent + 16497) {
                    // Overflow
                    xExponent = 0x7FFF;
                    xSignifier = 0;
                } else if (xExponent + msb + 16380 < yExponent) {
                    // Underflow
                    xExponent = 0;
                    xSignifier = 0;
                } else if (xExponent + msb + 16268 < yExponent) {
                    // Subnormal
                    if (xExponent + 16380 > yExponent)
                        xSignifier <<= xExponent + 16380 - yExponent;
                    else if (xExponent + 16380 < yExponent)
                        xSignifier >>= yExponent - xExponent - 16380;

                    xExponent = 0;
                } else {
                    // Normal
                    if (msb > 112) xSignifier >>= msb - 112;

                    xSignifier &= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

                    xExponent = xExponent + msb + 16269 - yExponent;
                }

                return
                    bytes16(
                        uint128(
                            uint128(
                                (x ^ y) & 0x80000000000000000000000000000000
                            ) |
                                (xExponent << 112) |
                                xSignifier
                        )
                    );
            }
        }
    }

    /**
     * Calculate -x.
     *
     * @param x quadruple precision number
     * @return quadruple precision number
     */
    function neg(bytes16 x) internal pure returns (bytes16) {
        unchecked {return x ^ 0x80000000000000000000000000000000;}
    }

    /**
     * Calculate |x|.
     *
     * @param x quadruple precision number
     * @return quadruple precision number
     */
    function abs(bytes16 x) internal pure returns (bytes16) {
        unchecked {return x & 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;}
    }

    /**
     * Calculate square root of x.  Return NaN on negative x excluding -0.
     *
     * @param x quadruple precision number
     * @return quadruple precision number
     */
    function sqrt(bytes16 x) internal pure returns (bytes16) {
        unchecked {
            if (uint128(x) > 0x80000000000000000000000000000000) return NaN;
            else {
                uint256 xExponent = (uint128(x) >> 112) & 0x7FFF;
                if (xExponent == 0x7FFF) return x;
                else {
                    uint256 xSignifier =
                        uint128(x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                    if (xExponent == 0) xExponent = 1;
                    else xSignifier |= 0x10000000000000000000000000000;

                    if (xSignifier == 0) return _POSITIVE_ZERO;

                    bool oddExponent = xExponent & 0x1 == 0;
                    xExponent = (xExponent + 16383) >> 1;

                    if (oddExponent) {
                        if (xSignifier >= 0x10000000000000000000000000000)
                            xSignifier <<= 113;
                        else {
                            uint256 msb = mostSignificantBit(xSignifier);
                            uint256 shift = (226 - msb) & 0xFE;
                            xSignifier <<= shift;
                            xExponent -= (shift - 112) >> 1;
                        }
                    } else {
                        if (xSignifier >= 0x10000000000000000000000000000)
                            xSignifier <<= 112;
                        else {
                            uint256 msb = mostSignificantBit(xSignifier);
                            uint256 shift = (225 - msb) & 0xFE;
                            xSignifier <<= shift;
                            xExponent -= (shift - 112) >> 1;
                        }
                    }

                    uint256 r = 0x10000000000000000000000000000;
                    r = (r + xSignifier / r) >> 1;
                    r = (r + xSignifier / r) >> 1;
                    r = (r + xSignifier / r) >> 1;
                    r = (r + xSignifier / r) >> 1;
                    r = (r + xSignifier / r) >> 1;
                    r = (r + xSignifier / r) >> 1;
                    r = (r + xSignifier / r) >> 1; // Seven iterations should be enough
                    uint256 r1 = xSignifier / r;
                    if (r1 < r) r = r1;

                    return
                        bytes16(
                            uint128(
                                (xExponent << 112) |
                                    (r & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF)
                            )
                        );
                }
            }
        }
    }

    /**
     * Calculate binary logarithm of x.  Return NaN on negative x excluding -0.
     *
     * @param x quadruple precision number
     * @return quadruple precision number
     */
    function log_2(bytes16 x) internal pure returns (bytes16) {
        unchecked {
            if (uint128(x) > 0x80000000000000000000000000000000) return NaN;
            else if (x == 0x3FFF0000000000000000000000000000)
                return _POSITIVE_ZERO;
            else {
                uint256 xExponent = (uint128(x) >> 112) & 0x7FFF;
                if (xExponent == 0x7FFF) return x;
                else {
                    uint256 xSignifier =
                        uint128(x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                    if (xExponent == 0) xExponent = 1;
                    else xSignifier |= 0x10000000000000000000000000000;

                    if (xSignifier == 0) return _NEGATIVE_INFINITY;

                    bool resultNegative;
                    uint256 resultExponent = 16495;
                    uint256 resultSignifier;

                    if (xExponent >= 0x3FFF) {
                        resultNegative = false;
                        resultSignifier = xExponent - 0x3FFF;
                        xSignifier <<= 15;
                    } else {
                        resultNegative = true;
                        if (xSignifier >= 0x10000000000000000000000000000) {
                            resultSignifier = 0x3FFE - xExponent;
                            xSignifier <<= 15;
                        } else {
                            uint256 msb = mostSignificantBit(xSignifier);
                            resultSignifier = 16493 - msb;
                            xSignifier <<= 127 - msb;
                        }
                    }

                    if (xSignifier == 0x80000000000000000000000000000000) {
                        if (resultNegative) resultSignifier += 1;
                        uint256 shift =
                            112 - mostSignificantBit(resultSignifier);
                        resultSignifier <<= shift;
                        resultExponent -= shift;
                    } else {
                        uint256 bb = resultNegative ? 1 : 0;
                        while (
                            resultSignifier < 0x10000000000000000000000000000
                        ) {
                            resultSignifier <<= 1;
                            resultExponent -= 1;

                            xSignifier *= xSignifier;
                            uint256 b = xSignifier >> 255;
                            resultSignifier += b ^ bb;
                            xSignifier >>= 127 + b;
                        }
                    }

                    return
                        bytes16(
                            uint128(
                                (
                                    resultNegative
                                        ? 0x80000000000000000000000000000000
                                        : 0
                                ) |
                                    (resultExponent << 112) |
                                    (resultSignifier &
                                        0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF)
                            )
                        );
                }
            }
        }
    }

    /**
     * Calculate natural logarithm of x.  Return NaN on negative x excluding -0.
     *
     * @param x quadruple precision number
     * @return quadruple precision number
     */
    function ln(bytes16 x) internal pure returns (bytes16) {
        unchecked {return mul(log_2(x), 0x3FFE62E42FEFA39EF35793C7673007E5);}
    }

    /**
     * Calculate 2^x.
     *
     * @param x quadruple precision number
     * @return quadruple precision number
     */
    function pow_2(bytes16 x) internal pure returns (bytes16) {
        unchecked {
            bool xNegative = uint128(x) > 0x80000000000000000000000000000000;
            uint256 xExponent = (uint128(x) >> 112) & 0x7FFF;
            uint256 xSignifier = uint128(x) & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

            if (xExponent == 0x7FFF && xSignifier != 0) return NaN;
            else if (xExponent > 16397)
                return xNegative ? _POSITIVE_ZERO : _POSITIVE_INFINITY;
            else if (xExponent < 16255)
                return 0x3FFF0000000000000000000000000000;
            else {
                if (xExponent == 0) xExponent = 1;
                else xSignifier |= 0x10000000000000000000000000000;

                if (xExponent > 16367) xSignifier <<= xExponent - 16367;
                else if (xExponent < 16367) xSignifier >>= 16367 - xExponent;

                if (
                    xNegative &&
                    xSignifier > 0x406E00000000000000000000000000000000
                ) return _POSITIVE_ZERO;

                if (
                    !xNegative &&
                    xSignifier > 0x3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
                ) return _POSITIVE_INFINITY;

                uint256 resultExponent = xSignifier >> 128;
                xSignifier &= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                if (xNegative && xSignifier != 0) {
                    xSignifier = ~xSignifier;
                    resultExponent += 1;
                }

                uint256 resultSignifier = 0x80000000000000000000000000000000;
                if (xSignifier & 0x80000000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x16A09E667F3BCC908B2FB1366EA957D3E) >>
                        128;
                if (xSignifier & 0x40000000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1306FE0A31B7152DE8D5A46305C85EDEC) >>
                        128;
                if (xSignifier & 0x20000000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1172B83C7D517ADCDF7C8C50EB14A791F) >>
                        128;
                if (xSignifier & 0x10000000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10B5586CF9890F6298B92B71842A98363) >>
                        128;
                if (xSignifier & 0x8000000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1059B0D31585743AE7C548EB68CA417FD) >>
                        128;
                if (xSignifier & 0x4000000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x102C9A3E778060EE6F7CACA4F7A29BDE8) >>
                        128;
                if (xSignifier & 0x2000000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10163DA9FB33356D84A66AE336DCDFA3F) >>
                        128;
                if (xSignifier & 0x1000000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100B1AFA5ABCBED6129AB13EC11DC9543) >>
                        128;
                if (xSignifier & 0x800000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10058C86DA1C09EA1FF19D294CF2F679B) >>
                        128;
                if (xSignifier & 0x400000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1002C605E2E8CEC506D21BFC89A23A00F) >>
                        128;
                if (xSignifier & 0x200000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100162F3904051FA128BCA9C55C31E5DF) >>
                        128;
                if (xSignifier & 0x100000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000B175EFFDC76BA38E31671CA939725) >>
                        128;
                if (xSignifier & 0x80000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100058BA01FB9F96D6CACD4B180917C3D) >>
                        128;
                if (xSignifier & 0x40000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10002C5CC37DA9491D0985C348C68E7B3) >>
                        128;
                if (xSignifier & 0x20000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000162E525EE054754457D5995292026) >>
                        128;
                if (xSignifier & 0x10000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000B17255775C040618BF4A4ADE83FC) >>
                        128;
                if (xSignifier & 0x8000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000058B91B5BC9AE2EED81E9B7D4CFAB) >>
                        128;
                if (xSignifier & 0x4000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100002C5C89D5EC6CA4D7C8ACC017B7C9) >>
                        128;
                if (xSignifier & 0x2000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000162E43F4F831060E02D839A9D16D) >>
                        128;
                if (xSignifier & 0x1000000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000B1721BCFC99D9F890EA06911763) >>
                        128;
                if (xSignifier & 0x800000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000058B90CF1E6D97F9CA14DBCC1628) >>
                        128;
                if (xSignifier & 0x400000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000002C5C863B73F016468F6BAC5CA2B) >>
                        128;
                if (xSignifier & 0x200000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000162E430E5A18F6119E3C02282A5) >>
                        128;
                if (xSignifier & 0x100000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000B1721835514B86E6D96EFD1BFE) >>
                        128;
                if (xSignifier & 0x80000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000058B90C0B48C6BE5DF846C5B2EF) >>
                        128;
                if (xSignifier & 0x40000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000002C5C8601CC6B9E94213C72737A) >>
                        128;
                if (xSignifier & 0x20000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000162E42FFF037DF38AA2B219F06) >>
                        128;
                if (xSignifier & 0x10000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000B17217FBA9C739AA5819F44F9) >>
                        128;
                if (xSignifier & 0x8000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000058B90BFCDEE5ACD3C1CEDC823) >>
                        128;
                if (xSignifier & 0x4000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000002C5C85FE31F35A6A30DA1BE50) >>
                        128;
                if (xSignifier & 0x2000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000162E42FF0999CE3541B9FFFCF) >>
                        128;
                if (xSignifier & 0x1000000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000B17217F80F4EF5AADDA45554) >>
                        128;
                if (xSignifier & 0x800000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000058B90BFBF8479BD5A81B51AD) >>
                        128;
                if (xSignifier & 0x400000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000002C5C85FDF84BD62AE30A74CC) >>
                        128;
                if (xSignifier & 0x200000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000162E42FEFB2FED257559BDAA) >>
                        128;
                if (xSignifier & 0x100000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000B17217F7D5A7716BBA4A9AE) >>
                        128;
                if (xSignifier & 0x80000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000058B90BFBE9DDBAC5E109CCE) >>
                        128;
                if (xSignifier & 0x40000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000002C5C85FDF4B15DE6F17EB0D) >>
                        128;
                if (xSignifier & 0x20000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000162E42FEFA494F1478FDE05) >>
                        128;
                if (xSignifier & 0x10000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000B17217F7D20CF927C8E94C) >>
                        128;
                if (xSignifier & 0x8000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000058B90BFBE8F71CB4E4B33D) >>
                        128;
                if (xSignifier & 0x4000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000002C5C85FDF477B662B26945) >>
                        128;
                if (xSignifier & 0x2000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000162E42FEFA3AE53369388C) >>
                        128;
                if (xSignifier & 0x1000000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000B17217F7D1D351A389D40) >>
                        128;
                if (xSignifier & 0x800000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000058B90BFBE8E8B2D3D4EDE) >>
                        128;
                if (xSignifier & 0x400000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000002C5C85FDF4741BEA6E77E) >>
                        128;
                if (xSignifier & 0x200000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000162E42FEFA39FE95583C2) >>
                        128;
                if (xSignifier & 0x100000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000B17217F7D1CFB72B45E1) >>
                        128;
                if (xSignifier & 0x80000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000058B90BFBE8E7CC35C3F0) >>
                        128;
                if (xSignifier & 0x40000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000002C5C85FDF473E242EA38) >>
                        128;
                if (xSignifier & 0x20000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000162E42FEFA39F02B772C) >>
                        128;
                if (xSignifier & 0x10000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000B17217F7D1CF7D83C1A) >>
                        128;
                if (xSignifier & 0x8000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000058B90BFBE8E7BDCBE2E) >>
                        128;
                if (xSignifier & 0x4000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000002C5C85FDF473DEA871F) >>
                        128;
                if (xSignifier & 0x2000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000162E42FEFA39EF44D91) >>
                        128;
                if (xSignifier & 0x1000000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000B17217F7D1CF79E949) >>
                        128;
                if (xSignifier & 0x800000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000058B90BFBE8E7BCE544) >>
                        128;
                if (xSignifier & 0x400000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000002C5C85FDF473DE6ECA) >>
                        128;
                if (xSignifier & 0x200000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000162E42FEFA39EF366F) >>
                        128;
                if (xSignifier & 0x100000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000B17217F7D1CF79AFA) >>
                        128;
                if (xSignifier & 0x80000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000058B90BFBE8E7BCD6D) >>
                        128;
                if (xSignifier & 0x40000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000002C5C85FDF473DE6B2) >>
                        128;
                if (xSignifier & 0x20000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000162E42FEFA39EF358) >>
                        128;
                if (xSignifier & 0x10000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000B17217F7D1CF79AB) >>
                        128;
                if (xSignifier & 0x8000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000058B90BFBE8E7BCD5) >>
                        128;
                if (xSignifier & 0x4000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000002C5C85FDF473DE6A) >>
                        128;
                if (xSignifier & 0x2000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000162E42FEFA39EF34) >>
                        128;
                if (xSignifier & 0x1000000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000B17217F7D1CF799) >>
                        128;
                if (xSignifier & 0x800000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000058B90BFBE8E7BCC) >>
                        128;
                if (xSignifier & 0x400000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000002C5C85FDF473DE5) >>
                        128;
                if (xSignifier & 0x200000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000162E42FEFA39EF2) >>
                        128;
                if (xSignifier & 0x100000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000B17217F7D1CF78) >>
                        128;
                if (xSignifier & 0x80000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000058B90BFBE8E7BB) >>
                        128;
                if (xSignifier & 0x40000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000002C5C85FDF473DD) >>
                        128;
                if (xSignifier & 0x20000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000162E42FEFA39EE) >>
                        128;
                if (xSignifier & 0x10000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000B17217F7D1CF6) >>
                        128;
                if (xSignifier & 0x8000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000058B90BFBE8E7A) >>
                        128;
                if (xSignifier & 0x4000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000002C5C85FDF473C) >>
                        128;
                if (xSignifier & 0x2000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000162E42FEFA39D) >>
                        128;
                if (xSignifier & 0x1000000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000B17217F7D1CE) >>
                        128;
                if (xSignifier & 0x800000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000058B90BFBE8E6) >>
                        128;
                if (xSignifier & 0x400000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000002C5C85FDF472) >>
                        128;
                if (xSignifier & 0x200000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000162E42FEFA38) >>
                        128;
                if (xSignifier & 0x100000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000000B17217F7D1B) >>
                        128;
                if (xSignifier & 0x80000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000058B90BFBE8D) >>
                        128;
                if (xSignifier & 0x40000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000002C5C85FDF46) >>
                        128;
                if (xSignifier & 0x20000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000000162E42FEFA2) >>
                        128;
                if (xSignifier & 0x10000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000000B17217F7D0) >>
                        128;
                if (xSignifier & 0x8000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000000058B90BFBE7) >>
                        128;
                if (xSignifier & 0x4000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000002C5C85FDF3) >>
                        128;
                if (xSignifier & 0x2000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000000162E42FEF9) >>
                        128;
                if (xSignifier & 0x1000000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000000B17217F7C) >>
                        128;
                if (xSignifier & 0x800000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000000058B90BFBD) >>
                        128;
                if (xSignifier & 0x400000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000000002C5C85FDE) >>
                        128;
                if (xSignifier & 0x200000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000000162E42FEE) >>
                        128;
                if (xSignifier & 0x100000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000000000B17217F6) >>
                        128;
                if (xSignifier & 0x80000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000000058B90BFA) >>
                        128;
                if (xSignifier & 0x40000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000000002C5C85FC) >>
                        128;
                if (xSignifier & 0x20000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000000000162E42FD) >>
                        128;
                if (xSignifier & 0x10000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000000000B17217E) >>
                        128;
                if (xSignifier & 0x8000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000000000058B90BE) >>
                        128;
                if (xSignifier & 0x4000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000000002C5C85E) >>
                        128;
                if (xSignifier & 0x2000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000000000162E42E) >>
                        128;
                if (xSignifier & 0x1000000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000000000B17216) >>
                        128;
                if (xSignifier & 0x800000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000000000058B90A) >>
                        128;
                if (xSignifier & 0x400000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000000000002C5C84) >>
                        128;
                if (xSignifier & 0x200000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000000000162E41) >>
                        128;
                if (xSignifier & 0x100000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000000000000B1720) >>
                        128;
                if (xSignifier & 0x80000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000000000058B8F) >>
                        128;
                if (xSignifier & 0x40000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000000000002C5C7) >>
                        128;
                if (xSignifier & 0x20000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000000000000162E3) >>
                        128;
                if (xSignifier & 0x10000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000000000000B171) >>
                        128;
                if (xSignifier & 0x8000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000000000000058B8) >>
                        128;
                if (xSignifier & 0x4000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000000000002C5B) >>
                        128;
                if (xSignifier & 0x2000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000000000000162D) >>
                        128;
                if (xSignifier & 0x1000 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000000000000B16) >>
                        128;
                if (xSignifier & 0x800 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000000000000058A) >>
                        128;
                if (xSignifier & 0x400 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000000000000002C4) >>
                        128;
                if (xSignifier & 0x200 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000000000000161) >>
                        128;
                if (xSignifier & 0x100 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x1000000000000000000000000000000B0) >>
                        128;
                if (xSignifier & 0x80 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000000000000057) >>
                        128;
                if (xSignifier & 0x40 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000000000000002B) >>
                        128;
                if (xSignifier & 0x20 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000000000000015) >>
                        128;
                if (xSignifier & 0x10 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x10000000000000000000000000000000A) >>
                        128;
                if (xSignifier & 0x8 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000000000000004) >>
                        128;
                if (xSignifier & 0x4 > 0)
                    resultSignifier =
                        (resultSignifier *
                            0x100000000000000000000000000000001) >>
                        128;

                if (!xNegative) {
                    resultSignifier =
                        (resultSignifier >> 15) &
                        0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                    resultExponent += 0x3FFF;
                } else if (resultExponent <= 0x3FFE) {
                    resultSignifier =
                        (resultSignifier >> 15) &
                        0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF;
                    resultExponent = 0x3FFF - resultExponent;
                } else {
                    resultSignifier =
                        resultSignifier >>
                        (resultExponent - 16367);
                    resultExponent = 0;
                }

                return
                    bytes16(uint128((resultExponent << 112) | resultSignifier));
            }
        }
    }

    /**
     * Calculate e^x.
     *
     * @param x quadruple precision number
     * @return quadruple precision number
     */
    function exp(bytes16 x) internal pure returns (bytes16) {
        unchecked {return pow_2(mul(x, 0x3FFF71547652B82FE1777D0FFDA0D23A));}
    }

    /**
     * Get index of the most significant non-zero bit in binary representation of
     * x.  Reverts if x is zero.
     *
     * @return index of the most significant non-zero bit in binary representation
     *         of x
     */
    function mostSignificantBit(uint256 x) private pure returns (uint256) {
        unchecked {
            require(x > 0);

            uint256 result = 0;

            if (x >= 0x100000000000000000000000000000000) {
                x >>= 128;
                result += 128;
            }
            if (x >= 0x10000000000000000) {
                x >>= 64;
                result += 64;
            }
            if (x >= 0x100000000) {
                x >>= 32;
                result += 32;
            }
            if (x >= 0x10000) {
                x >>= 16;
                result += 16;
            }
            if (x >= 0x100) {
                x >>= 8;
                result += 8;
            }
            if (x >= 0x10) {
                x >>= 4;
                result += 4;
            }
            if (x >= 0x4) {
                x >>= 2;
                result += 2;
            }
            if (x >= 0x2) result += 1; // No need to shift x anymore

            return result;
        }
    }
}

// File: @openzeppelin/contracts/access/AccessControl.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

import "../utils/Context.sol";
import "../utils/Strings.sol";
import "../utils/introspection/ERC165.sol";

/**
 * @dev External interface of AccessControl declared to support ERC165 detection.
 */
interface IAccessControl {
    function hasRole(bytes32 role, address account) external view returns (bool);
    function getRoleAdmin(bytes32 role) external view returns (bytes32);
    function grantRole(bytes32 role, address account) external;
    function revokeRole(bytes32 role, address account) external;
    function renounceRole(bytes32 role, address account) external;
}

/**
 * @dev Contract module that allows children to implement role-based access
 * control mechanisms. This is a lightweight version that doesn't allow enumerating role
 * members except through off-chain means by accessing the contract event logs. Some
 * applications may benefit from on-chain enumerability, for those cases see
 * {AccessControlEnumerable}.
 *
 * Roles are referred to by their `bytes32` identifier. These should be exposed
 * in the external API and be unique. The best way to achieve this is by
 * using `public constant` hash digests:
 *
 * ```
 * bytes32 public constant MY_ROLE = keccak256("MY_ROLE");
 * ```
 *
 * Roles can be used to represent a set of permissions. To restrict access to a
 * function call, use {hasRole}:
 *
 * ```
 * function foo() public {
 *     require(hasRole(MY_ROLE, msg.sender));
 *     ...
 * }
 * ```
 *
 * Roles can be granted and revoked dynamically via the {grantRole} and
 * {revokeRole} functions. Each role has an associated admin role, and only
 * accounts that have a role's admin role can call {grantRole} and {revokeRole}.
 *
 * By default, the admin role for all roles is `DEFAULT_ADMIN_ROLE`, which means
 * that only accounts with this role will be able to grant or revoke other
 * roles. More complex role relationships can be created by using
 * {_setRoleAdmin}.
 *
 * WARNING: The `DEFAULT_ADMIN_ROLE` is also its own admin: it has permission to
 * grant and revoke this role. Extra precautions should be taken to secure
 * accounts that have been granted it.
 */
abstract contract AccessControl is Context, IAccessControl, ERC165 {
    struct RoleData {
        mapping (address => bool) members;
        bytes32 adminRole;
    }

    mapping (bytes32 => RoleData) private _roles;

    bytes32 public constant DEFAULT_ADMIN_ROLE = 0x00;

    /**
     * @dev Emitted when `newAdminRole` is set as ``role``'s admin role, replacing `previousAdminRole`
     *
     * `DEFAULT_ADMIN_ROLE` is the starting admin for all roles, despite
     * {RoleAdminChanged} not being emitted signaling this.
     *
     * _Available since v3.1._
     */
    event RoleAdminChanged(bytes32 indexed role, bytes32 indexed previousAdminRole, bytes32 indexed newAdminRole);

    /**
     * @dev Emitted when `account` is granted `role`.
     *
     * `sender` is the account that originated the contract call, an admin role
     * bearer except when using {_setupRole}.
     */
    event RoleGranted(bytes32 indexed role, address indexed account, address indexed sender);

    /**
     * @dev Emitted when `account` is revoked `role`.
     *
     * `sender` is the account that originated the contract call:
     *   - if using `revokeRole`, it is the admin role bearer
     *   - if using `renounceRole`, it is the role bearer (i.e. `account`)
     */
    event RoleRevoked(bytes32 indexed role, address indexed account, address indexed sender);

    /**
     * @dev Modifier that checks that an account has a specific role. Reverts
     * with a standardized message including the required role.
     *
     * The format of the revert reason is given by the following regular expression:
     *
     *  /^AccessControl: account (0x[0-9a-f]{20}) is missing role (0x[0-9a-f]{32})$/
     *
     * _Available since v4.1._
     */
    modifier onlyRole(bytes32 role) {
        _checkRole(role, _msgSender());
        _;
    }

    /**
     * @dev See {IERC165-supportsInterface}.
     */
    function supportsInterface(bytes4 interfaceId) public view virtual override returns (bool) {
        return interfaceId == type(IAccessControl).interfaceId
            || super.supportsInterface(interfaceId);
    }

    /**
     * @dev Returns `true` if `account` has been granted `role`.
     */
    function hasRole(bytes32 role, address account) public view override returns (bool) {
        return _roles[role].members[account];
    }

    /**
     * @dev Revert with a standard message if `account` is missing `role`.
     *
     * The format of the revert reason is given by the following regular expression:
     *
     *  /^AccessControl: account (0x[0-9a-f]{20}) is missing role (0x[0-9a-f]{32})$/
     */
    function _checkRole(bytes32 role, address account) internal view {
        if(!hasRole(role, account)) {
            revert(string(abi.encodePacked(
                "AccessControl: account ",
                Strings.toHexString(uint160(account), 20),
                " is missing role ",
                Strings.toHexString(uint256(role), 32)
            )));
        }
    }

    /**
     * @dev Returns the admin role that controls `role`. See {grantRole} and
     * {revokeRole}.
     *
     * To change a role's admin, use {_setRoleAdmin}.
     */
    function getRoleAdmin(bytes32 role) public view override returns (bytes32) {
        return _roles[role].adminRole;
    }

    /**
     * @dev Grants `role` to `account`.
     *
     * If `account` had not been already granted `role`, emits a {RoleGranted}
     * event.
     *
     * Requirements:
     *
     * - the caller must have ``role``'s admin role.
     */
    function grantRole(bytes32 role, address account) public virtual override onlyRole(getRoleAdmin(role)) {
        _grantRole(role, account);
    }

    /**
     * @dev Revokes `role` from `account`.
     *
     * If `account` had been granted `role`, emits a {RoleRevoked} event.
     *
     * Requirements:
     *
     * - the caller must have ``role``'s admin role.
     */
    function revokeRole(bytes32 role, address account) public virtual override onlyRole(getRoleAdmin(role)) {
        _revokeRole(role, account);
    }

    /**
     * @dev Revokes `role` from the calling account.
     *
     * Roles are often managed via {grantRole} and {revokeRole}: this function's
     * purpose is to provide a mechanism for accounts to lose their privileges
     * if they are compromised (such as when a trusted device is misplaced).
     *
     * If the calling account had been granted `role`, emits a {RoleRevoked}
     * event.
     *
     * Requirements:
     *
     * - the caller must be `account`.
     */
    function renounceRole(bytes32 role, address account) public virtual override {
        require(account == _msgSender(), "AccessControl: can only renounce roles for self");

        _revokeRole(role, account);
    }

    /**
     * @dev Grants `role` to `account`.
     *
     * If `account` had not been already granted `role`, emits a {RoleGranted}
     * event. Note that unlike {grantRole}, this function doesn't perform any
     * checks on the calling account.
     *
     * [WARNING]
     * ====
     * This function should only be called from the constructor when setting
     * up the initial roles for the system.
     *
     * Using this function in any other way is effectively circumventing the admin
     * system imposed by {AccessControl}.
     * ====
     */
    function _setupRole(bytes32 role, address account) internal virtual {
        _grantRole(role, account);
    }

    /**
     * @dev Sets `adminRole` as ``role``'s admin role.
     *
     * Emits a {RoleAdminChanged} event.
     */
    function _setRoleAdmin(bytes32 role, bytes32 adminRole) internal virtual {
        emit RoleAdminChanged(role, getRoleAdmin(role), adminRole);
        _roles[role].adminRole = adminRole;
    }

    function _grantRole(bytes32 role, address account) private {
        if (!hasRole(role, account)) {
            _roles[role].members[account] = true;
            emit RoleGranted(role, account, _msgSender());
        }
    }

    function _revokeRole(bytes32 role, address account) private {
        if (hasRole(role, account)) {
            _roles[role].members[account] = false;
            emit RoleRevoked(role, account, _msgSender());
        }
    }
}

// File: @openzeppelin/contracts/token/ERC20/ERC20.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

import "./IERC20.sol";
import "./extensions/IERC20Metadata.sol";
import "../../utils/Context.sol";

/**
 * @dev Implementation of the {IERC20} interface.
 *
 * This implementation is agnostic to the way tokens are created. This means
 * that a supply mechanism has to be added in a derived contract using {_mint}.
 * For a generic mechanism see {ERC20PresetMinterPauser}.
 *
 * TIP: For a detailed writeup see our guide
 * https://forum.zeppelin.solutions/t/how-to-implement-erc20-supply-mechanisms/226[How
 * to implement supply mechanisms].
 *
 * We have followed general OpenZeppelin guidelines: functions revert instead
 * of returning `false` on failure. This behavior is nonetheless conventional
 * and does not conflict with the expectations of ERC20 applications.
 *
 * Additionally, an {Approval} event is emitted on calls to {transferFrom}.
 * This allows applications to reconstruct the allowance for all accounts just
 * by listening to said events. Other implementations of the EIP may not emit
 * these events, as it isn't required by the specification.
 *
 * Finally, the non-standard {decreaseAllowance} and {increaseAllowance}
 * functions have been added to mitigate the well-known issues around setting
 * allowances. See {IERC20-approve}.
 */
contract ERC20 is Context, IERC20, IERC20Metadata {
    mapping (address => uint256) private _balances;

    mapping (address => mapping (address => uint256)) private _allowances;

    uint256 private _totalSupply;

    string private _name;
    string private _symbol;

    /**
     * @dev Sets the values for {name} and {symbol}.
     *
     * The defaut value of {decimals} is 18. To select a different value for
     * {decimals} you should overload it.
     *
     * All two of these values are immutable: they can only be set once during
     * construction.
     */
    constructor (string memory name_, string memory symbol_) {
        _name = name_;
        _symbol = symbol_;
    }

    /**
     * @dev Returns the name of the token.
     */
    function name() public view virtual override returns (string memory) {
        return _name;
    }

    /**
     * @dev Returns the symbol of the token, usually a shorter version of the
     * name.
     */
    function symbol() public view virtual override returns (string memory) {
        return _symbol;
    }

    /**
     * @dev Returns the number of decimals used to get its user representation.
     * For example, if `decimals` equals `2`, a balance of `505` tokens should
     * be displayed to a user as `5,05` (`505 / 10 ** 2`).
     *
     * Tokens usually opt for a value of 18, imitating the relationship between
     * Ether and Wei. This is the value {ERC20} uses, unless this function is
     * overridden;
     *
     * NOTE: This information is only used for _display_ purposes: it in
     * no way affects any of the arithmetic of the contract, including
     * {IERC20-balanceOf} and {IERC20-transfer}.
     */
    function decimals() public view virtual override returns (uint8) {
        return 18;
    }

    /**
     * @dev See {IERC20-totalSupply}.
     */
    function totalSupply() public view virtual override returns (uint256) {
        return _totalSupply;
    }

    /**
     * @dev See {IERC20-balanceOf}.
     */
    function balanceOf(address account) public view virtual override returns (uint256) {
        return _balances[account];
    }

    /**
     * @dev See {IERC20-transfer}.
     *
     * Requirements:
     *
     * - `recipient` cannot be the zero address.
     * - the caller must have a balance of at least `amount`.
     */
    function transfer(address recipient, uint256 amount) public virtual override returns (bool) {
        _transfer(_msgSender(), recipient, amount);
        return true;
    }

    /**
     * @dev See {IERC20-allowance}.
     */
    function allowance(address owner, address spender) public view virtual override returns (uint256) {
        return _allowances[owner][spender];
    }

    /**
     * @dev See {IERC20-approve}.
     *
     * Requirements:
     *
     * - `spender` cannot be the zero address.
     */
    function approve(address spender, uint256 amount) public virtual override returns (bool) {
        _approve(_msgSender(), spender, amount);
        return true;
    }

    /**
     * @dev See {IERC20-transferFrom}.
     *
     * Emits an {Approval} event indicating the updated allowance. This is not
     * required by the EIP. See the note at the beginning of {ERC20}.
     *
     * Requirements:
     *
     * - `sender` and `recipient` cannot be the zero address.
     * - `sender` must have a balance of at least `amount`.
     * - the caller must have allowance for ``sender``'s tokens of at least
     * `amount`.
     */
    function transferFrom(address sender, address recipient, uint256 amount) public virtual override returns (bool) {
        _transfer(sender, recipient, amount);

        uint256 currentAllowance = _allowances[sender][_msgSender()];
        require(currentAllowance >= amount, "ERC20: transfer amount exceeds allowance");
        _approve(sender, _msgSender(), currentAllowance - amount);

        return true;
    }

    /**
     * @dev Atomically increases the allowance granted to `spender` by the caller.
     *
     * This is an alternative to {approve} that can be used as a mitigation for
     * problems described in {IERC20-approve}.
     *
     * Emits an {Approval} event indicating the updated allowance.
     *
     * Requirements:
     *
     * - `spender` cannot be the zero address.
     */
    function increaseAllowance(address spender, uint256 addedValue) public virtual returns (bool) {
        _approve(_msgSender(), spender, _allowances[_msgSender()][spender] + addedValue);
        return true;
    }

    /**
     * @dev Atomically decreases the allowance granted to `spender` by the caller.
     *
     * This is an alternative to {approve} that can be used as a mitigation for
     * problems described in {IERC20-approve}.
     *
     * Emits an {Approval} event indicating the updated allowance.
     *
     * Requirements:
     *
     * - `spender` cannot be the zero address.
     * - `spender` must have allowance for the caller of at least
     * `subtractedValue`.
     */
    function decreaseAllowance(address spender, uint256 subtractedValue) public virtual returns (bool) {
        uint256 currentAllowance = _allowances[_msgSender()][spender];
        require(currentAllowance >= subtractedValue, "ERC20: decreased allowance below zero");
        _approve(_msgSender(), spender, currentAllowance - subtractedValue);

        return true;
    }

    /**
     * @dev Moves tokens `amount` from `sender` to `recipient`.
     *
     * This is internal function is equivalent to {transfer}, and can be used to
     * e.g. implement automatic token fees, slashing mechanisms, etc.
     *
     * Emits a {Transfer} event.
     *
     * Requirements:
     *
     * - `sender` cannot be the zero address.
     * - `recipient` cannot be the zero address.
     * - `sender` must have a balance of at least `amount`.
     */
    function _transfer(address sender, address recipient, uint256 amount) internal virtual {
        require(sender != address(0), "ERC20: transfer from the zero address");
        require(recipient != address(0), "ERC20: transfer to the zero address");

        _beforeTokenTransfer(sender, recipient, amount);

        uint256 senderBalance = _balances[sender];
        require(senderBalance >= amount, "ERC20: transfer amount exceeds balance");
        _balances[sender] = senderBalance - amount;
        _balances[recipient] += amount;

        emit Transfer(sender, recipient, amount);
    }

    /** @dev Creates `amount` tokens and assigns them to `account`, increasing
     * the total supply.
     *
     * Emits a {Transfer} event with `from` set to the zero address.
     *
     * Requirements:
     *
     * - `to` cannot be the zero address.
     */
    function _mint(address account, uint256 amount) internal virtual {
        require(account != address(0), "ERC20: mint to the zero address");

        _beforeTokenTransfer(address(0), account, amount);

        _totalSupply += amount;
        _balances[account] += amount;
        emit Transfer(address(0), account, amount);
    }

    /**
     * @dev Destroys `amount` tokens from `account`, reducing the
     * total supply.
     *
     * Emits a {Transfer} event with `to` set to the zero address.
     *
     * Requirements:
     *
     * - `account` cannot be the zero address.
     * - `account` must have at least `amount` tokens.
     */
    function _burn(address account, uint256 amount) internal virtual {
        require(account != address(0), "ERC20: burn from the zero address");

        _beforeTokenTransfer(account, address(0), amount);

        uint256 accountBalance = _balances[account];
        require(accountBalance >= amount, "ERC20: burn amount exceeds balance");
        _balances[account] = accountBalance - amount;
        _totalSupply -= amount;

        emit Transfer(account, address(0), amount);
    }

    /**
     * @dev Sets `amount` as the allowance of `spender` over the `owner` s tokens.
     *
     * This internal function is equivalent to `approve`, and can be used to
     * e.g. set automatic allowances for certain subsystems, etc.
     *
     * Emits an {Approval} event.
     *
     * Requirements:
     *
     * - `owner` cannot be the zero address.
     * - `spender` cannot be the zero address.
     */
    function _approve(address owner, address spender, uint256 amount) internal virtual {
        require(owner != address(0), "ERC20: approve from the zero address");
        require(spender != address(0), "ERC20: approve to the zero address");

        _allowances[owner][spender] = amount;
        emit Approval(owner, spender, amount);
    }

    /**
     * @dev Hook that is called before any transfer of tokens. This includes
     * minting and burning.
     *
     * Calling conditions:
     *
     * - when `from` and `to` are both non-zero, `amount` of ``from``'s tokens
     * will be to transferred to `to`.
     * - when `from` is zero, `amount` tokens will be minted for `to`.
     * - when `to` is zero, `amount` of ``from``'s tokens will be burned.
     * - `from` and `to` are never both zero.
     *
     * To learn more about hooks, head to xref:ROOT:extending-contracts.adoc#using-hooks[Using Hooks].
     */
    function _beforeTokenTransfer(address from, address to, uint256 amount) internal virtual { }
}

// File: @openzeppelin/contracts/token/ERC20/IERC20.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

/**
 * @dev Interface of the ERC20 standard as defined in the EIP.
 */
interface IERC20 {
    /**
     * @dev Returns the amount of tokens in existence.
     */
    function totalSupply() external view returns (uint256);

    /**
     * @dev Returns the amount of tokens owned by `account`.
     */
    function balanceOf(address account) external view returns (uint256);

    /**
     * @dev Moves `amount` tokens from the caller's account to `recipient`.
     *
     * Returns a boolean value indicating whether the operation succeeded.
     *
     * Emits a {Transfer} event.
     */
    function transfer(address recipient, uint256 amount) external returns (bool);

    /**
     * @dev Returns the remaining number of tokens that `spender` will be
     * allowed to spend on behalf of `owner` through {transferFrom}. This is
     * zero by default.
     *
     * This value changes when {approve} or {transferFrom} are called.
     */
    function allowance(address owner, address spender) external view returns (uint256);

    /**
     * @dev Sets `amount` as the allowance of `spender` over the caller's tokens.
     *
     * Returns a boolean value indicating whether the operation succeeded.
     *
     * IMPORTANT: Beware that changing an allowance with this method brings the risk
     * that someone may use both the old and the new allowance by unfortunate
     * transaction ordering. One possible solution to mitigate this race
     * condition is to first reduce the spender's allowance to 0 and set the
     * desired value afterwards:
     * https://github.com/ethereum/EIPs/issues/20#issuecomment-263524729
     *
     * Emits an {Approval} event.
     */
    function approve(address spender, uint256 amount) external returns (bool);

    /**
     * @dev Moves `amount` tokens from `sender` to `recipient` using the
     * allowance mechanism. `amount` is then deducted from the caller's
     * allowance.
     *
     * Returns a boolean value indicating whether the operation succeeded.
     *
     * Emits a {Transfer} event.
     */
    function transferFrom(address sender, address recipient, uint256 amount) external returns (bool);

    /**
     * @dev Emitted when `value` tokens are moved from one account (`from`) to
     * another (`to`).
     *
     * Note that `value` may be zero.
     */
    event Transfer(address indexed from, address indexed to, uint256 value);

    /**
     * @dev Emitted when the allowance of a `spender` for an `owner` is set by
     * a call to {approve}. `value` is the new allowance.
     */
    event Approval(address indexed owner, address indexed spender, uint256 value);
}

// File: @openzeppelin/contracts/token/ERC20/utils/SafeERC20.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

import "../IERC20.sol";
import "../../../utils/Address.sol";

/**
 * @title SafeERC20
 * @dev Wrappers around ERC20 operations that throw on failure (when the token
 * contract returns false). Tokens that return no value (and instead revert or
 * throw on failure) are also supported, non-reverting calls are assumed to be
 * successful.
 * To use this library you can add a `using SafeERC20 for IERC20;` statement to your contract,
 * which allows you to call the safe operations as `token.safeTransfer(...)`, etc.
 */
library SafeERC20 {
    using Address for address;

    function safeTransfer(IERC20 token, address to, uint256 value) internal {
        _callOptionalReturn(token, abi.encodeWithSelector(token.transfer.selector, to, value));
    }

    function safeTransferFrom(IERC20 token, address from, address to, uint256 value) internal {
        _callOptionalReturn(token, abi.encodeWithSelector(token.transferFrom.selector, from, to, value));
    }

    /**
     * @dev Deprecated. This function has issues similar to the ones found in
     * {IERC20-approve}, and its usage is discouraged.
     *
     * Whenever possible, use {safeIncreaseAllowance} and
     * {safeDecreaseAllowance} instead.
     */
    function safeApprove(IERC20 token, address spender, uint256 value) internal {
        // safeApprove should only be called when setting an initial allowance,
        // or when resetting it to zero. To increase and decrease it, use
        // 'safeIncreaseAllowance' and 'safeDecreaseAllowance'
        // solhint-disable-next-line max-line-length
        require((value == 0) || (token.allowance(address(this), spender) == 0),
            "SafeERC20: approve from non-zero to non-zero allowance"
        );
        _callOptionalReturn(token, abi.encodeWithSelector(token.approve.selector, spender, value));
    }

    function safeIncreaseAllowance(IERC20 token, address spender, uint256 value) internal {
        uint256 newAllowance = token.allowance(address(this), spender) + value;
        _callOptionalReturn(token, abi.encodeWithSelector(token.approve.selector, spender, newAllowance));
    }

    function safeDecreaseAllowance(IERC20 token, address spender, uint256 value) internal {
        unchecked {
            uint256 oldAllowance = token.allowance(address(this), spender);
            require(oldAllowance >= value, "SafeERC20: decreased allowance below zero");
            uint256 newAllowance = oldAllowance - value;
            _callOptionalReturn(token, abi.encodeWithSelector(token.approve.selector, spender, newAllowance));
        }
    }

    /**
     * @dev Imitates a Solidity high-level call (i.e. a regular function call to a contract), relaxing the requirement
     * on the return value: the return value is optional (but if data is returned, it must not be false).
     * @param token The token targeted by the call.
     * @param data The call data (encoded using abi.encode or one of its variants).
     */
    function _callOptionalReturn(IERC20 token, bytes memory data) private {
        // We need to perform a low level call here, to bypass Solidity's return data size checking mechanism, since
        // we're implementing it ourselves. We use {Address.functionCall} to perform this call, which verifies that
        // the target address contains contract code and also asserts for success in the low-level call.

        bytes memory returndata = address(token).functionCall(data, "SafeERC20: low-level call failed");
        if (returndata.length > 0) { // Return data is optional
            // solhint-disable-next-line max-line-length
            require(abi.decode(returndata, (bool)), "SafeERC20: ERC20 operation did not succeed");
        }
    }
}

// File: contracts/interfaces/IUbiquityAlgorithmicDollar.sol



// SPDX-License-Identifier: Apache-2.0
pragma solidity ^0.8.3;

import "./IERC20Ubiquity.sol";

/// @title UAD stablecoin interface
/// @author Ubiquity Algorithmic Dollar
interface IUbiquityAlgorithmicDollar is IERC20Ubiquity {
    event IncentiveContractUpdate(
        address indexed _incentivized,
        address indexed _incentiveContract
    );

    function setIncentiveContract(address account, address incentive) external;

    function incentiveContract(address account) external view returns (address);
}

// File: contracts/interfaces/ICurveFactory.sol



// SPDX-License-Identifier: MIT
// !! THIS FILE WAS AUTOGENERATED BY abi-to-sol. SEE BELOW FOR SOURCE. !!
pragma solidity ^0.8.3;

interface ICurveFactory {
    event BasePoolAdded(address base_pool, address implementat);
    event MetaPoolDeployed(
        address coin,
        address base_pool,
        uint256 A,
        uint256 fee,
        address deployer
    );

    function find_pool_for_coins(address _from, address _to)
        external
        view
        returns (address);

    function find_pool_for_coins(
        address _from,
        address _to,
        uint256 i
    ) external view returns (address);

    function get_n_coins(address _pool)
        external
        view
        returns (uint256, uint256);

    function get_coins(address _pool) external view returns (address[2] memory);

    function get_underlying_coins(address _pool)
        external
        view
        returns (address[8] memory);

    function get_decimals(address _pool)
        external
        view
        returns (uint256[2] memory);

    function get_underlying_decimals(address _pool)
        external
        view
        returns (uint256[8] memory);

    function get_rates(address _pool) external view returns (uint256[2] memory);

    function get_balances(address _pool)
        external
        view
        returns (uint256[2] memory);

    function get_underlying_balances(address _pool)
        external
        view
        returns (uint256[8] memory);

    function get_A(address _pool) external view returns (uint256);

    function get_fees(address _pool) external view returns (uint256, uint256);

    function get_admin_balances(address _pool)
        external
        view
        returns (uint256[2] memory);

    function get_coin_indices(
        address _pool,
        address _from,
        address _to
    )
        external
        view
        returns (
            int128,
            int128,
            bool
        );

    function add_base_pool(
        address _base_pool,
        address _metapool_implementation,
        address _fee_receiver
    ) external;

    function deploy_metapool(
        address _base_pool,
        string memory _name,
        string memory _symbol,
        address _coin,
        uint256 _A,
        uint256 _fee
    ) external returns (address);

    function commit_transfer_ownership(address addr) external;

    function accept_transfer_ownership() external;

    function set_fee_receiver(address _base_pool, address _fee_receiver)
        external;

    function convert_fees() external returns (bool);

    function admin() external view returns (address);

    function future_admin() external view returns (address);

    function pool_list(uint256 arg0) external view returns (address);

    function pool_count() external view returns (uint256);

    function base_pool_list(uint256 arg0) external view returns (address);

    function base_pool_count() external view returns (uint256);

    function fee_receiver(address arg0) external view returns (address);
}

// File: contracts/interfaces/IMetaPool.sol



// SPDX-License-Identifier: UNLICENSED
// !! THIS FILE WAS AUTOGENERATED BY abi-to-sol. SEE BELOW FOR SOURCE. !!
pragma solidity ^0.8.3;

interface IMetaPool {
    event Transfer(
        address indexed sender,
        address indexed receiver,
        uint256 value
    );
    event Approval(
        address indexed owner,
        address indexed spender,
        uint256 value
    );
    event TokenExchange(
        address indexed buyer,
        int128 sold_id,
        uint256 tokens_sold,
        int128 bought_id,
        uint256 tokens_bought
    );
    event TokenExchangeUnderlying(
        address indexed buyer,
        int128 sold_id,
        uint256 tokens_sold,
        int128 bought_id,
        uint256 tokens_bought
    );
    event AddLiquidity(
        address indexed provider,
        uint256[2] token_amounts,
        uint256[2] fees,
        uint256 invariant,
        uint256 token_supply
    );
    event RemoveLiquidity(
        address indexed provider,
        uint256[2] token_amounts,
        uint256[2] fees,
        uint256 token_supply
    );
    event RemoveLiquidityOne(
        address indexed provider,
        uint256 token_amount,
        uint256 coin_amount,
        uint256 token_supply
    );
    event RemoveLiquidityImbalance(
        address indexed provider,
        uint256[2] token_amounts,
        uint256[2] fees,
        uint256 invariant,
        uint256 token_supply
    );
    event CommitNewAdmin(uint256 indexed deadline, address indexed admin);
    event NewAdmin(address indexed admin);
    event CommitNewFee(
        uint256 indexed deadline,
        uint256 fee,
        uint256 admin_fee
    );
    event NewFee(uint256 fee, uint256 admin_fee);
    event RampA(
        uint256 old_A,
        uint256 new_A,
        uint256 initial_time,
        uint256 future_time
    );
    event StopRampA(uint256 A, uint256 t);

    function initialize(
        string memory _name,
        string memory _symbol,
        address _coin,
        uint256 _decimals,
        uint256 _A,
        uint256 _fee,
        address _admin
    ) external;

    function decimals() external view returns (uint256);

    function transfer(address _to, uint256 _value) external returns (bool);

    function transferFrom(
        address _from,
        address _to,
        uint256 _value
    ) external returns (bool);

    function approve(address _spender, uint256 _value) external returns (bool);

    function get_previous_balances() external view returns (uint256[2] memory);

    function get_balances() external view returns (uint256[2] memory);

    function get_twap_balances(
        uint256[2] memory _first_balances,
        uint256[2] memory _last_balances,
        uint256 _time_elapsed
    ) external view returns (uint256[2] memory);

    function get_price_cumulative_last()
        external
        view
        returns (uint256[2] memory);

    function admin_fee() external view returns (uint256);

    function A() external view returns (uint256);

    function A_precise() external view returns (uint256);

    function get_virtual_price() external view returns (uint256);

    function calc_token_amount(uint256[2] memory _amounts, bool _is_deposit)
        external
        view
        returns (uint256);

    function calc_token_amount(
        uint256[2] memory _amounts,
        bool _is_deposit,
        bool _previous
    ) external view returns (uint256);

    function add_liquidity(uint256[2] memory _amounts, uint256 _min_mint_amount)
        external
        returns (uint256);

    function add_liquidity(
        uint256[2] memory _amounts,
        uint256 _min_mint_amount,
        address _receiver
    ) external returns (uint256);

    function get_dy(
        int128 i,
        int128 j,
        uint256 dx
    ) external view returns (uint256);

    function get_dy(
        int128 i,
        int128 j,
        uint256 dx,
        uint256[2] memory _balances
    ) external view returns (uint256);

    function get_dy_underlying(
        int128 i,
        int128 j,
        uint256 dx
    ) external view returns (uint256);

    function get_dy_underlying(
        int128 i,
        int128 j,
        uint256 dx,
        uint256[2] memory _balances
    ) external view returns (uint256);

    function exchange(
        int128 i,
        int128 j,
        uint256 dx,
        uint256 min_dy
    ) external returns (uint256);

    function exchange(
        int128 i,
        int128 j,
        uint256 dx,
        uint256 min_dy,
        address _receiver
    ) external returns (uint256);

    function exchange_underlying(
        int128 i,
        int128 j,
        uint256 dx,
        uint256 min_dy
    ) external returns (uint256);

    function exchange_underlying(
        int128 i,
        int128 j,
        uint256 dx,
        uint256 min_dy,
        address _receiver
    ) external returns (uint256);

    function remove_liquidity(
        uint256 _burn_amount,
        uint256[2] memory _min_amounts
    ) external returns (uint256[2] memory);

    function remove_liquidity(
        uint256 _burn_amount,
        uint256[2] memory _min_amounts,
        address _receiver
    ) external returns (uint256[2] memory);

    function remove_liquidity_imbalance(
        uint256[2] memory _amounts,
        uint256 _max_burn_amount
    ) external returns (uint256);

    function remove_liquidity_imbalance(
        uint256[2] memory _amounts,
        uint256 _max_burn_amount,
        address _receiver
    ) external returns (uint256);

    function calc_withdraw_one_coin(uint256 _burn_amount, int128 i)
        external
        view
        returns (uint256);

    function calc_withdraw_one_coin(
        uint256 _burn_amount,
        int128 i,
        bool _previous
    ) external view returns (uint256);

    function remove_liquidity_one_coin(
        uint256 _burn_amount,
        int128 i,
        uint256 _min_received
    ) external returns (uint256);

    function remove_liquidity_one_coin(
        uint256 _burn_amount,
        int128 i,
        uint256 _min_received,
        address _receiver
    ) external returns (uint256);

    function ramp_A(uint256 _future_A, uint256 _future_time) external;

    function stop_ramp_A() external;

    function admin_balances(uint256 i) external view returns (uint256);

    function withdraw_admin_fees() external;

    function admin() external view returns (address);

    function coins(uint256 arg0) external view returns (address);

    function balances(uint256 arg0) external view returns (uint256);

    function fee() external view returns (uint256);

    function block_timestamp_last() external view returns (uint256);

    function initial_A() external view returns (uint256);

    function future_A() external view returns (uint256);

    function initial_A_time() external view returns (uint256);

    function future_A_time() external view returns (uint256);

    function name() external view returns (string memory);

    function symbol() external view returns (string memory);

    function balanceOf(address arg0) external view returns (uint256);

    function allowance(address arg0, address arg1)
        external
        view
        returns (uint256);

    function totalSupply() external view returns (uint256);
}

// File: @openzeppelin/contracts/utils/Context.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

/*
 * @dev Provides information about the current execution context, including the
 * sender of the transaction and its data. While these are generally available
 * via msg.sender and msg.data, they should not be accessed in such a direct
 * manner, since when dealing with meta-transactions the account sending and
 * paying for execution may not be the actual sender (as far as an application
 * is concerned).
 *
 * This contract is only required for intermediate, library-like contracts.
 */
abstract contract Context {
    function _msgSender() internal view virtual returns (address) {
        return msg.sender;
    }

    function _msgData() internal view virtual returns (bytes calldata) {
        this; // silence state mutability warning without generating bytecode - see https://github.com/ethereum/solidity/issues/2691
        return msg.data;
    }
}

// File: @openzeppelin/contracts/utils/Strings.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

/**
 * @dev String operations.
 */
library Strings {
    bytes16 private constant alphabet = "0123456789abcdef";

    /**
     * @dev Converts a `uint256` to its ASCII `string` decimal representation.
     */
    function toString(uint256 value) internal pure returns (string memory) {
        // Inspired by OraclizeAPI's implementation - MIT licence
        // https://github.com/oraclize/ethereum-api/blob/b42146b063c7d6ee1358846c198246239e9360e8/oraclizeAPI_0.4.25.sol

        if (value == 0) {
            return "0";
        }
        uint256 temp = value;
        uint256 digits;
        while (temp != 0) {
            digits++;
            temp /= 10;
        }
        bytes memory buffer = new bytes(digits);
        while (value != 0) {
            digits -= 1;
            buffer[digits] = bytes1(uint8(48 + uint256(value % 10)));
            value /= 10;
        }
        return string(buffer);
    }

    /**
     * @dev Converts a `uint256` to its ASCII `string` hexadecimal representation.
     */
    function toHexString(uint256 value) internal pure returns (string memory) {
        if (value == 0) {
            return "0x00";
        }
        uint256 temp = value;
        uint256 length = 0;
        while (temp != 0) {
            length++;
            temp >>= 8;
        }
        return toHexString(value, length);
    }

    /**
     * @dev Converts a `uint256` to its ASCII `string` hexadecimal representation with fixed length.
     */
    function toHexString(uint256 value, uint256 length) internal pure returns (string memory) {
        bytes memory buffer = new bytes(2 * length + 2);
        buffer[0] = "0";
        buffer[1] = "x";
        for (uint256 i = 2 * length + 1; i > 1; --i) {
            buffer[i] = alphabet[value & 0xf];
            value >>= 4;
        }
        require(value == 0, "Strings: hex length insufficient");
        return string(buffer);
    }

}

// File: @openzeppelin/contracts/utils/introspection/ERC165.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

import "./IERC165.sol";

/**
 * @dev Implementation of the {IERC165} interface.
 *
 * Contracts that want to implement ERC165 should inherit from this contract and override {supportsInterface} to check
 * for the additional interface id that will be supported. For example:
 *
 * ```solidity
 * function supportsInterface(bytes4 interfaceId) public view virtual override returns (bool) {
 *     return interfaceId == type(MyInterface).interfaceId || super.supportsInterface(interfaceId);
 * }
 * ```
 *
 * Alternatively, {ERC165Storage} provides an easier to use but more expensive implementation.
 */
abstract contract ERC165 is IERC165 {
    /**
     * @dev See {IERC165-supportsInterface}.
     */
    function supportsInterface(bytes4 interfaceId) public view virtual override returns (bool) {
        return interfaceId == type(IERC165).interfaceId;
    }
}

// File: @openzeppelin/contracts/utils/introspection/IERC165.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

/**
 * @dev Interface of the ERC165 standard, as defined in the
 * https://eips.ethereum.org/EIPS/eip-165[EIP].
 *
 * Implementers can declare support of contract interfaces, which can then be
 * queried by others ({ERC165Checker}).
 *
 * For an implementation, see {ERC165}.
 */
interface IERC165 {
    /**
     * @dev Returns true if this contract implements the interface defined by
     * `interfaceId`. See the corresponding
     * https://eips.ethereum.org/EIPS/eip-165#how-interfaces-are-identified[EIP section]
     * to learn more about how these ids are created.
     *
     * This function call must use less than 30 000 gas.
     */
    function supportsInterface(bytes4 interfaceId) external view returns (bool);
}

// File: @openzeppelin/contracts/token/ERC20/extensions/IERC20Metadata.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

import "../IERC20.sol";

/**
 * @dev Interface for the optional metadata functions from the ERC20 standard.
 *
 * _Available since v4.1._
 */
interface IERC20Metadata is IERC20 {
    /**
     * @dev Returns the name of the token.
     */
    function name() external view returns (string memory);

    /**
     * @dev Returns the symbol of the token.
     */
    function symbol() external view returns (string memory);

    /**
     * @dev Returns the decimals places of the token.
     */
    function decimals() external view returns (uint8);
}

// File: @openzeppelin/contracts/utils/Address.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

/**
 * @dev Collection of functions related to the address type
 */
library Address {
    /**
     * @dev Returns true if `account` is a contract.
     *
     * [IMPORTANT]
     * ====
     * It is unsafe to assume that an address for which this function returns
     * false is an externally-owned account (EOA) and not a contract.
     *
     * Among others, `isContract` will return false for the following
     * types of addresses:
     *
     *  - an externally-owned account
     *  - a contract in construction
     *  - an address where a contract will be created
     *  - an address where a contract lived, but was destroyed
     * ====
     */
    function isContract(address account) internal view returns (bool) {
        // This method relies on extcodesize, which returns 0 for contracts in
        // construction, since the code is only stored at the end of the
        // constructor execution.

        uint256 size;
        // solhint-disable-next-line no-inline-assembly
        assembly { size := extcodesize(account) }
        return size > 0;
    }

    /**
     * @dev Replacement for Solidity's `transfer`: sends `amount` wei to
     * `recipient`, forwarding all available gas and reverting on errors.
     *
     * https://eips.ethereum.org/EIPS/eip-1884[EIP1884] increases the gas cost
     * of certain opcodes, possibly making contracts go over the 2300 gas limit
     * imposed by `transfer`, making them unable to receive funds via
     * `transfer`. {sendValue} removes this limitation.
     *
     * https://diligence.consensys.net/posts/2019/09/stop-using-soliditys-transfer-now/[Learn more].
     *
     * IMPORTANT: because control is transferred to `recipient`, care must be
     * taken to not create reentrancy vulnerabilities. Consider using
     * {ReentrancyGuard} or the
     * https://solidity.readthedocs.io/en/v0.5.11/security-considerations.html#use-the-checks-effects-interactions-pattern[checks-effects-interactions pattern].
     */
    function sendValue(address payable recipient, uint256 amount) internal {
        require(address(this).balance >= amount, "Address: insufficient balance");

        // solhint-disable-next-line avoid-low-level-calls, avoid-call-value
        (bool success, ) = recipient.call{ value: amount }("");
        require(success, "Address: unable to send value, recipient may have reverted");
    }

    /**
     * @dev Performs a Solidity function call using a low level `call`. A
     * plain`call` is an unsafe replacement for a function call: use this
     * function instead.
     *
     * If `target` reverts with a revert reason, it is bubbled up by this
     * function (like regular Solidity function calls).
     *
     * Returns the raw returned data. To convert to the expected return value,
     * use https://solidity.readthedocs.io/en/latest/units-and-global-variables.html?highlight=abi.decode#abi-encoding-and-decoding-functions[`abi.decode`].
     *
     * Requirements:
     *
     * - `target` must be a contract.
     * - calling `target` with `data` must not revert.
     *
     * _Available since v3.1._
     */
    function functionCall(address target, bytes memory data) internal returns (bytes memory) {
      return functionCall(target, data, "Address: low-level call failed");
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`], but with
     * `errorMessage` as a fallback revert reason when `target` reverts.
     *
     * _Available since v3.1._
     */
    function functionCall(address target, bytes memory data, string memory errorMessage) internal returns (bytes memory) {
        return functionCallWithValue(target, data, 0, errorMessage);
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`],
     * but also transferring `value` wei to `target`.
     *
     * Requirements:
     *
     * - the calling contract must have an ETH balance of at least `value`.
     * - the called Solidity function must be `payable`.
     *
     * _Available since v3.1._
     */
    function functionCallWithValue(address target, bytes memory data, uint256 value) internal returns (bytes memory) {
        return functionCallWithValue(target, data, value, "Address: low-level call with value failed");
    }

    /**
     * @dev Same as {xref-Address-functionCallWithValue-address-bytes-uint256-}[`functionCallWithValue`], but
     * with `errorMessage` as a fallback revert reason when `target` reverts.
     *
     * _Available since v3.1._
     */
    function functionCallWithValue(address target, bytes memory data, uint256 value, string memory errorMessage) internal returns (bytes memory) {
        require(address(this).balance >= value, "Address: insufficient balance for call");
        require(isContract(target), "Address: call to non-contract");

        // solhint-disable-next-line avoid-low-level-calls
        (bool success, bytes memory returndata) = target.call{ value: value }(data);
        return _verifyCallResult(success, returndata, errorMessage);
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`],
     * but performing a static call.
     *
     * _Available since v3.3._
     */
    function functionStaticCall(address target, bytes memory data) internal view returns (bytes memory) {
        return functionStaticCall(target, data, "Address: low-level static call failed");
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-string-}[`functionCall`],
     * but performing a static call.
     *
     * _Available since v3.3._
     */
    function functionStaticCall(address target, bytes memory data, string memory errorMessage) internal view returns (bytes memory) {
        require(isContract(target), "Address: static call to non-contract");

        // solhint-disable-next-line avoid-low-level-calls
        (bool success, bytes memory returndata) = target.staticcall(data);
        return _verifyCallResult(success, returndata, errorMessage);
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`],
     * but performing a delegate call.
     *
     * _Available since v3.4._
     */
    function functionDelegateCall(address target, bytes memory data) internal returns (bytes memory) {
        return functionDelegateCall(target, data, "Address: low-level delegate call failed");
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-string-}[`functionCall`],
     * but performing a delegate call.
     *
     * _Available since v3.4._
     */
    function functionDelegateCall(address target, bytes memory data, string memory errorMessage) internal returns (bytes memory) {
        require(isContract(target), "Address: delegate call to non-contract");

        // solhint-disable-next-line avoid-low-level-calls
        (bool success, bytes memory returndata) = target.delegatecall(data);
        return _verifyCallResult(success, returndata, errorMessage);
    }

    function _verifyCallResult(bool success, bytes memory returndata, string memory errorMessage) private pure returns(bytes memory) {
        if (success) {
            return returndata;
        } else {
            // Look for revert reason and bubble it up if present
            if (returndata.length > 0) {
                // The easiest way to bubble the revert reason is using memory via assembly

                // solhint-disable-next-line no-inline-assembly
                assembly {
                    let returndata_size := mload(returndata)
                    revert(add(32, returndata), returndata_size)
                }
            } else {
                revert(errorMessage);
            }
        }
    }
}

// File: contracts/interfaces/IERC20Ubiquity.sol



// SPDX-License-Identifier: Apache-2.0
pragma solidity ^0.8.3;

import "@openzeppelin/contracts/token/ERC20/IERC20.sol";

/// @title ERC20 Ubiquiti preset interface
/// @author Ubiquity Algorithmic Dollar
interface IERC20Ubiquity is IERC20 {
    // ----------- Events -----------
    event Minting(
        address indexed _to,
        address indexed _minter,
        uint256 _amount
    );

    event Burning(address indexed _burned, uint256 _amount);

    // ----------- State changing api -----------
    function burn(uint256 amount) external;

    function permit(
        address owner,
        address spender,
        uint256 value,
        uint256 deadline,
        uint8 v,
        bytes32 r,
        bytes32 s
    ) external;

    // ----------- Burner only state changing api -----------
    function burnFrom(address account, uint256 amount) external;

    // ----------- Minter only state changing api -----------
    function mint(address account, uint256 amount) external;
}

// File: contracts/ERC20Ubiquity.sol



// SPDX-License-Identifier: MIT
pragma solidity ^0.8.3;

import "@openzeppelin/contracts/token/ERC20/ERC20.sol";
import "@openzeppelin/contracts/token/ERC20/extensions/ERC20Burnable.sol";
import "@openzeppelin/contracts/token/ERC20/extensions/ERC20Pausable.sol";
import "./UbiquityAlgorithmicDollarManager.sol";
import "./interfaces/IERC20Ubiquity.sol";

/// @title ERC20 Ubiquity preset
/// @author Ubiquity Algorithmic Dollar
/// @notice ERC20 with :
/// - ERC20 minter, burner and pauser
/// - draft-ERC20 permit
/// - Ubiquity Manager access control
contract ERC20Ubiquity is IERC20Ubiquity, ERC20, ERC20Burnable, ERC20Pausable {
    UbiquityAlgorithmicDollarManager public manager;

    // solhint-disable-next-line var-name-mixedcase
    bytes32 public DOMAIN_SEPARATOR;
    // keccak256("Permit(address owner,address spender,
    //                   uint256 value,uint256 nonce,uint256 deadline)");
    bytes32 public constant PERMIT_TYPEHASH =
        0x6e71edae12b1b97f4d1f60370fef10105fa2faae0126114a169c64845d6126c9;
    mapping(address => uint256) public nonces;
    string private _tokenName;
    string private _symbol;

    // ----------- Modifiers -----------
    modifier onlyMinter() {
        require(
            manager.hasRole(manager.UBQ_MINTER_ROLE(), msg.sender),
            "Governance token: not minter"
        );
        _;
    }

    modifier onlyBurner() {
        require(
            manager.hasRole(manager.UBQ_BURNER_ROLE(), msg.sender),
            "Governance token: not burner"
        );
        _;
    }

    modifier onlyPauser() {
        require(
            manager.hasRole(manager.PAUSER_ROLE(), msg.sender),
            "Governance token: not pauser"
        );
        _;
    }

    modifier onlyAdmin() {
        require(
            manager.hasRole(manager.DEFAULT_ADMIN_ROLE(), msg.sender),
            "ERC20: deployer must be manager admin"
        );
        _;
    }

    constructor(
        address _manager,
        string memory name_,
        string memory symbol_
    ) ERC20(name_, symbol_) {
        _tokenName = name_;
        _symbol = symbol_;
        manager = UbiquityAlgorithmicDollarManager(_manager);
        // sender must be UbiquityAlgorithmicDollarManager roleAdmin
        // because he will get the admin, minter and pauser role on uAD and we want to
        // manage all permissions through the manager
        require(
            manager.hasRole(manager.DEFAULT_ADMIN_ROLE(), msg.sender),
            "ERC20: deployer must be manager admin"
        );
        uint256 chainId;
        // solhint-disable-next-line no-inline-assembly
        assembly {
            chainId := chainid()
        }

        DOMAIN_SEPARATOR = keccak256(
            abi.encode(
                keccak256(
                    // solhint-disable-next-line max-line-length
                    "EIP712Domain(string name,string version,uint256 chainId,address verifyingContract)"
                ),
                keccak256(bytes(name())),
                keccak256(bytes("1")),
                chainId,
                address(this)
            )
        );
    }

    /// @notice setSymbol update token symbol
    /// @param newSymbol new token symbol
    function setSymbol(string memory newSymbol) external onlyAdmin {
        _symbol = newSymbol;
    }

    /// @notice setName update token name
    /// @param newName new token name
    function setName(string memory newName) external onlyAdmin {
        _tokenName = newName;
    }

    /// @notice permit spending of uAD. owner has signed a message allowing
    ///         spender to transfer up to amount uAD
    /// @param owner the uAD holder
    /// @param spender the approved operator
    /// @param value the amount approved
    /// @param deadline the deadline after which the approval is no longer valid
    function permit(
        address owner,
        address spender,
        uint256 value,
        uint256 deadline,
        uint8 v,
        bytes32 r,
        bytes32 s
    ) external override {
        // solhint-disable-next-line not-rely-on-time
        require(deadline >= block.timestamp, "Dollar: EXPIRED");
        bytes32 digest =
            keccak256(
                abi.encodePacked(
                    "\x19\x01",
                    DOMAIN_SEPARATOR,
                    keccak256(
                        abi.encode(
                            PERMIT_TYPEHASH,
                            owner,
                            spender,
                            value,
                            nonces[owner]++,
                            deadline
                        )
                    )
                )
            );
        address recoveredAddress = ecrecover(digest, v, r, s);
        require(
            recoveredAddress != address(0) && recoveredAddress == owner,
            "Dollar: INVALID_SIGNATURE"
        );
        _approve(owner, spender, value);
    }

    /// @notice burn UAD tokens from caller
    /// @param amount the amount to burn
    function burn(uint256 amount)
        public
        override(ERC20Burnable, IERC20Ubiquity)
        whenNotPaused
    {
        super.burn(amount);
        emit Burning(msg.sender, amount);
    }

    /// @notice burn uAD tokens from specified account
    /// @param account the account to burn from
    /// @param amount the amount to burn
    function burnFrom(address account, uint256 amount)
        public
        override(ERC20Burnable, IERC20Ubiquity)
        onlyBurner
        whenNotPaused // to suppress ? if BURNER_ROLE should do it even paused ?
    {
        _burn(account, amount);
        emit Burning(account, amount);
    }

    // @dev Creates `amount` new tokens for `to`.
    function mint(address to, uint256 amount)
        public
        override
        onlyMinter
        whenNotPaused
    {
        _mint(to, amount);
        emit Minting(to, msg.sender, amount);
    }

    // @dev Pauses all token transfers.
    function pause() public onlyPauser {
        _pause();
    }

    // @dev Unpauses all token transfers.
    function unpause() public onlyPauser {
        _unpause();
    }

    /**
     * @dev Returns the name of the token.
     */
    function name() public view override(ERC20) returns (string memory) {
        return _tokenName;
    }

    /**
     * @dev Returns the symbol of the token, usually a shorter version of the
     * name.
     */
    function symbol() public view override(ERC20) returns (string memory) {
        return _symbol;
    }

    function _beforeTokenTransfer(
        address from,
        address to,
        uint256 amount
    ) internal virtual override(ERC20, ERC20Pausable) {
        super._beforeTokenTransfer(from, to, amount);
    }

    function _transfer(
        address sender,
        address recipient,
        uint256 amount
    ) internal virtual override whenNotPaused {
        super._transfer(sender, recipient, amount);
    }
}

// File: @openzeppelin/contracts/token/ERC20/extensions/ERC20Burnable.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

import "../ERC20.sol";
import "../../../utils/Context.sol";

/**
 * @dev Extension of {ERC20} that allows token holders to destroy both their own
 * tokens and those that they have an allowance for, in a way that can be
 * recognized off-chain (via event analysis).
 */
abstract contract ERC20Burnable is Context, ERC20 {
    /**
     * @dev Destroys `amount` tokens from the caller.
     *
     * See {ERC20-_burn}.
     */
    function burn(uint256 amount) public virtual {
        _burn(_msgSender(), amount);
    }

    /**
     * @dev Destroys `amount` tokens from `account`, deducting from the caller's
     * allowance.
     *
     * See {ERC20-_burn} and {ERC20-allowance}.
     *
     * Requirements:
     *
     * - the caller must have allowance for ``accounts``'s tokens of at least
     * `amount`.
     */
    function burnFrom(address account, uint256 amount) public virtual {
        uint256 currentAllowance = allowance(account, _msgSender());
        require(currentAllowance >= amount, "ERC20: burn amount exceeds allowance");
        _approve(account, _msgSender(), currentAllowance - amount);
        _burn(account, amount);
    }
}

// File: @openzeppelin/contracts/token/ERC20/extensions/ERC20Pausable.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

import "../ERC20.sol";
import "../../../security/Pausable.sol";

/**
 * @dev ERC20 token with pausable token transfers, minting and burning.
 *
 * Useful for scenarios such as preventing trades until the end of an evaluation
 * period, or having an emergency switch for freezing all token transfers in the
 * event of a large bug.
 */
abstract contract ERC20Pausable is ERC20, Pausable {
    /**
     * @dev See {ERC20-_beforeTokenTransfer}.
     *
     * Requirements:
     *
     * - the contract must not be paused.
     */
    function _beforeTokenTransfer(address from, address to, uint256 amount) internal virtual override {
        super._beforeTokenTransfer(from, to, amount);

        require(!paused(), "ERC20Pausable: token transfer while paused");
    }
}

// File: @openzeppelin/contracts/security/Pausable.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

import "../utils/Context.sol";

/**
 * @dev Contract module which allows children to implement an emergency stop
 * mechanism that can be triggered by an authorized account.
 *
 * This module is used through inheritance. It will make available the
 * modifiers `whenNotPaused` and `whenPaused`, which can be applied to
 * the functions of your contract. Note that they will not be pausable by
 * simply including this module, only once the modifiers are put in place.
 */
abstract contract Pausable is Context {
    /**
     * @dev Emitted when the pause is triggered by `account`.
     */
    event Paused(address account);

    /**
     * @dev Emitted when the pause is lifted by `account`.
     */
    event Unpaused(address account);

    bool private _paused;

    /**
     * @dev Initializes the contract in unpaused state.
     */
    constructor () {
        _paused = false;
    }

    /**
     * @dev Returns true if the contract is paused, and false otherwise.
     */
    function paused() public view virtual returns (bool) {
        return _paused;
    }

    /**
     * @dev Modifier to make a function callable only when the contract is not paused.
     *
     * Requirements:
     *
     * - The contract must not be paused.
     */
    modifier whenNotPaused() {
        require(!paused(), "Pausable: paused");
        _;
    }

    /**
     * @dev Modifier to make a function callable only when the contract is paused.
     *
     * Requirements:
     *
     * - The contract must be paused.
     */
    modifier whenPaused() {
        require(paused(), "Pausable: not paused");
        _;
    }

    /**
     * @dev Triggers stopped state.
     *
     * Requirements:
     *
     * - The contract must not be paused.
     */
    function _pause() internal virtual whenNotPaused {
        _paused = true;
        emit Paused(_msgSender());
    }

    /**
     * @dev Returns to normal state.
     *
     * Requirements:
     *
     * - The contract must be paused.
     */
    function _unpause() internal virtual whenPaused {
        _paused = false;
        emit Unpaused(_msgSender());
    }
}